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Question 958413: Hello, I've been struggling with forming equations from word problems. I love math but word problems are my weakness, I've been stuck here trying to figure out the equations for days. My heads already hurting, please help me understand.
The word problem is: the sum of three numbers is 50. The second number is three times the first number, and the third number is twice the second number. What are the numbers?
(I know how to solve equations, i just don't know how to form equations using word problems. )
Answer by MathTherapy(10806)

(Show Source):

You can put this solution on YOUR website!

Hello, I've been struggling with forming equations from word problems. I love math but word problems are my weakness, I've been stuck
here trying to figure out the equations for days. My heads already hurting, please help me understand.
The word problem is: the sum of three numbers is 50. The second number is three times the first number, and the third number is twice
the second number. What are the numbers?

(I know how to solve equations, i just don't know how to form equations using word problems. )
*************************************************
Two other "respondents" assigned 3 different variables to the 3 numbers. I'd suggest that only ONE (1) variable be used.

Let's name the FIRST number, F
As ".....The second number is three times the first number," the 2nd number is: 3F
And, as "....the third number is twice the second number," the THIRD number is: 2(3F) = 6F
Finally, since, "the sum of three numbers is 50," we get: F + 3F + 6F = 50
10F = 50
First number, or
Second number: 3(5) = 15
Third number: 6(5) = 30

It's great NOT to be able to deal with 3 different variables, which can prove to be a HEADACHE, sometimes! Maybe that's
what brought on yours!

You can do the CHECK!!

Question 1210576: Finish the sequence 3. 6. 18 ___
which of these numbers is correct
84
60
72
30
Found 2 solutions by josgarithmetic, KMST:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
Guessing, based on factors or 2's and 3's

3, 2*3, 2*3*3, next can be 2*2*3*3=36

Not enough terms for a clear pattern

"Next" should be 72.
72=2*3*2*3*2
which uses only 2's and 3's.

Answer by KMST(5345)   (Show Source):
You can put this solution on YOUR website!
We start with and we add to the right to get

Question 1210575: Finish the sequence 3. 6. 18 ___
which of these numbers is correct
84
60
72
30
Answer by math_tutor2020(3835)   (Show Source):
You can put this solution on YOUR website!

This sequence could be geometric.
To find out, divide adjacent terms
term2/term1 = 6/3 = 2
term3/term2 = 18/6 = 3
The ratios we got were 2 and 3.
We do not get the same ratio each time, so the sequence isn't geometric.

But the pattern could be: multiply by 2, multiply by 3, multiply by 4, etc
If so, then
18*4 = 72
could be the next term.

Unfortunately sequence problems like this are very flawed.
Check out this similar problem
https://www.algebra.com/algebra/homework/Sequences-and-series/Sequences-and-series.faq.question.1195799.html
On that link I try to find the next few terms in the sequence 1,2,4,...
It turns out there are at least 3 different possible answers for that question. There may be infinitely many answers.
Without knowing the actual rule to "3, 6, 18, ..." there's no way of knowing 100% what the next term could be.

Question 1210552: Pls help me and solve this sequence:
The first and the third terms of a G.P are 5 and 80 respectively. find the 4th term
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

The 3rd term is the 1st term, multiplied by the common ratio twice:

Solve to find the common ratio r.

or

The 4th term is the 3rd term, multiplied by the common ratio one more time.

Since there are two possible values for the common ratio, there are two possibilities for the 4th term.

Since the problem asks us to find "the" 4th term, perhaps the question is at an elementary level and they expect us to use only positive numbers. In that case, the 4th term is .

But another possible value for the 4th term is .

ANSWERS:
possibly 320
or, more correctly, either 320 or -320

Question 1210536: Find the sum of first 100 terms
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
```python
n = 100
sum_100 = n * (n + 1) // 2
print(f"{sum_100=}")

```
```text
sum_100=5050

```
To find the sum of the first 100 terms of the arithmetic series , we can use the formula for the sum of the first natural numbers:
### Calculation:
Given :
**The sum of the first 100 terms is 5,050.**

Question 1204431: Determine whether 280 is a term in the G.P 1,3,9,27…
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
Determine whether 280 is a term in the G.P 1,3,9,27…
~~~~~~~~~~~~~~~~~~~~~~~~~~

Good question !

The solution is very simple: each term of the GP is a degree of 3;
so, in particular, starting from the second term, each term is a multiple of 3.

From the other side, the number 280 is NOT a multiple of 3 (just because
the sum of its digits, 2+8+0 = 10 is not a multiple of 3).

So, the number 280 is not a member of the given G.P.         ANSWER.

Solved.

Question 1130236: Lauren is planning to raise her batting average. When she set up her weekly practice schedule, her batting average was 250. Lauren intends to increase her batting average by 5 each week. Find a pattern and write a formula that will give her expected batting average after n weeks.
a1 =
d=
Using the formula, how many weeks will Lauren need to practice if she expects to earn a batting average of 300?
Found 2 solutions by n2, ikleyn:
Answer by n2(79)   (Show Source):
You can put this solution on YOUR website!
.

Starting from November 2025 and during December 2025 I worked on checking solutions
provided by @mananth at this forum www.algebra.com .

For now, Jan.06, 2026, I reviewed/checked about 2*17*31 = 1054 problems, solved
by @mananth at this web-site (problems from 2390..9419 to 10320..10349 inclusively
of the list https://www.algebra.com/tutors/your-answers.mpl?userid=mananth ).

Of them, I found about 99 problems solved incorrectly.
Below is the list of these problems, solved incorrectly.
For each of these problems, my correct solution is placed at the same spot/link.

I started this my checking job from some of the problems, and after checking approximately 70 problems,
I started noticing the repeating patterns in problems and in their solutions.
Quite often, these solutions were incorrect, and the errors were of the kind a live person would not
make them. It led me to the idea that the solution were actually produced not by a living person,
but by a computer code, instead. So, it was a very early version of the Artificial Intelligence.

After getting this observation, two ideas led me.

First idea was, of course, to clean this forum from erroneous solutions.
Second idea was to learn how far this artificial intelligence may lead in producing wrong and right solution.
So, this file is my report on this issue.

I collected here only wrong solutions (wrong from a Math point of view).
There were also cases, when the solutions were led to correct answers, but were wrong
from an educative/pedagogical point of view. Then suggestions "wrong/correct" become more individualized/personalized,
and I did not include such cases in this my file. So, only incorrect solutions are collected/listed here.

From about 1000 reviewed problems, the number of incorrectly solved is about 100 (for now).
In other words, about 10% of all problems were solved incorrectly by @mananth (by his version of AI).

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How long after the second runner started did the second runner overtake the first runner? https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.433975.html 433981 10 gallons of a 15.0% alcohol solution are to be mixed with a 28.0% alcohol solution to make a 20.0% alcohol solution. How many gallons of a 28.0% alcohol must be used? How many gallons of a 20.0% alcohol solution are made? https://www.algebra.com/algebra/homework/word/mixtures/Mixture_Word_Problems.faq.question.433981.html 434087 Solve this equation by sustitution x=8-4y 2x-3y=13 https://www.algebra.com/algebra/homework/coordinate/Linear-systems.faq.question.434087.html 431703 One plane traveled 800 miles in the same time a second plane traveled 1,000 miles. The rate of the second plane was 50 mi/h faster than the rate of the first plane. How fast did each plane fly? https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.431703.html 430695 write and equation of the line containing the points (3,-1) and is parallel to 8x-7y=9 https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.430695.html 430837 x^2+8x+15>=0 https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.430837.html 431238 which equation represents a line that passes through the points (2,3) and (-1, -3)? https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.431238.html 431339 how do u find the midpoint of the line segment of (10,-10),(9,-9)? https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.431339.html 430008 A developer needs $80,000 to buy land. He is able to borrow the money at 7% compounded quarterly. How much interest will be paid on the loan if it is paid off in 5 years? (To the nearest dollar, do not include the $) https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.430008.html 428476 Flying from Tokyo to London is approximately 6175 miles. On the way to London from Tokyo(against the wind) the flight took 13 hours. the return flight (with the wind) took 9.88 hours. Find the speed of the plane in still air and the speed of the wind current. https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.428476.html 430019 (80) Could anyone help me with the following? What would be the semiannual interest payment on a $24,000 Proctor and Gamble 10-year bond at 6.875%? (round to the nearest dollar) and... What would be total amount of interest earned on a $24,000 Proctor and Gamble 10-year bond at 6.875% over the entire life of the bond? Thank you in advance!! https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.430019.html 430373 Can anyone help me with the following? If you deposit $2500 in an account paying 4.5% interest compounded semiannually, how much will be in the account after 8 years? (Round to the nearest cent) Thank you!! https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.430373.html 418623 If the sum of the lengths of the edges of a cube is 48 inches, the volume of the cube is? https://www.algebra.com/algebra/homework/word/geometry/Geometry_Word_Problems.faq.question.418623.html 419077 Dave Horn invested half of his money at 5%, one-third of his money at4%, and the rest at 3.5%. If his total annual investment income is $530, how much had he invested? https://www.algebra.com/algebra/homework/word/finance/Money_Word_Problems.faq.question.419077.html 419129 A glue company needs to make some glue tat it can sell at $120 per barrel. It wants to use 150 barrels of glue worth $100 per barrel, along with glue worth $150 per barrel and glue worth $190 per barrel. It must use the same number of barrels of $150 and $190 glue. How much of the $150 and $190 glue will be needed? How many Barrels of $120 glue will be produced? https://www.algebra.com/algebra/homework/coordinate/word/Linear_Equations_And_Systems_Word_Problems.faq.question.419129.html 419040 The current of a river is 2 miles per hour. A boat travels to a point 8 miles upstream and back again in 3 hours. What is the speed of the boat in still water? https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.419040.html 419037 The speed of a boat in still water is 5 miles per hour. If the boat travels 3 miles downstream in the same amount of time it takes to travel 1.5 miles upstream, what is the speed of the current? https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.419037.html 419685 Hi tutor, i do not get this question what so ever. It his a problem solving, algebraic question. I really need help. It would be greatly appreciated, thank you so much, may god bless you. Q : 500 tickets for a football game were sold and the total receipts were $105. Some of the tickets sold for $15 and the rest sold for 25cents. Find the number of each sold. ( i am required too provide a equation/formula , write a table i already have that set up : i have it as ticket 1 & ticket two & value, number and amount. & i also would greatly appreciate if you show the work as too how you got the answers, so i can learn from it. Thank you so much. https://www.algebra.com/algebra/homework/word/coins/Word_Problems_With_Coins.faq.question.419685.html 420019 what is the slope-intercept form that contains the points (3,7) and (3,5) https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.420019.html 420867 If I have a 20 qt radiator containing a 80% antifreeze solution. How much of the solution should I drain and replace with pure water to get a solution that is 50% antifreeze? https://www.algebra.com/algebra/homework/word/evaluation/Evaluation_Word_Problems.faq.question.420867.html 421830 (90) -4x-2y=-8 y=-2x+4 algebra.com/algebra/homework/Expressions-with-variables/Expressions-with-variables.faq.question.421830.html 421840 Solve each system of equations using the substitution method. 4x - y = 32y = -2 x + 70 https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Matrices-and-determiminant.faq.question.421840.html 422281 I'm trying to solve when a man knows he has lost 12.4 percent of his weight, and now weighs 249.6. What was his starting weight? https://www.algebra.com/algebra/homework/proportions/Proportions.faq.question.422281.html 423336 A liter of orange fruit drink contains 22% orange juice. How many millimeters of orange juice must be added to produce a mixture containing 50% orange juice? https://www.algebra.com/algebra/homework/percentage/percentage.faq.question.423336.html 423677 what is the equation of a line that passes through (-5, 1) and is parallel to y = x + 4 ? please show steps to help me understand https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.423677.html 423206 solving linear systems using substitution, what is 4x-7y=10 y=x-7 https://www.algebra.com/algebra/homework/Systems-of-equations/Systems-of-equations.faq.question.423206.html 423288 An airplane flew for 3 hours with a tail wind of 18 kilometers per hour. The return flight against the wind took 4 hours. Find the rate of the plane in still air. https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.423288.html 424727 Children's tickets to a movie cost $4. Adult tickets cost $7. If 275 tickets were sold for a total cost of $1174, how many of each type were sold? https://www.algebra.com/algebra/homework/coordinate/word/Linear_Equations_And_Systems_Word_Problems.faq.question.424727.html 426400 It takes a freight train 3 hours more to travel 280 miles than it takes an express train to travel 200 miles. The rate of the express is 10 miles per hour faster the the rate of the freight train. Fine the rates of both trains. https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.426400.html 425874 (99) a lion devours a sheep in 4 hours. a leopard devours a sheep in 5 hours. a bear devours a sheep in 6 hours. how long will it take the three animals eating simultaneously to consume a single sheep? https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.425874.html

/\/\/\/\/\/\/\/\/\/\/\/

You may ask me - ok, @mananth is a computer code, or AI - what does it change ?

Ok, below are my thoughts and my suggestions about it.

(1) First, 10% wrong solutions is too much for an AI. You may say "ok, in 20 years it will be perfect". Probably, it can be true, but I tell you about what I see now. By the way, my feeling is that good AI acceptable level for wrong solutions should be at most 1:1000, or 0.1%. (2) Second, it can be perfect in 20 years under one necessary condition: the best minds in Math solving and writing and the best computer programming specialists will work on it starting from now and during 20 years. What does it mean and why ? - Raising and educating a specialist in solving Math, writing Math and explaining Math takes 10 - 15 - 20 years to get a professional level, when a person can speak, write and teach professionally. (And, usually, the fingers of one hand are enough to count such specialists of a national level in each country and in each generation). If you make such experiments with the novices, it will take 10-15-20 years for them to get a professional level before they will learn to speak/write at the professional level, so it is a WRONG way to assign novices for key positions for such an activity. It is the second conclusion from my arguments: the writers and the content creators must be professionals from the very beginning. Neither @mananth, nor @CPhill are such professionals. (3) Third, the computer codes which I see in the posts by @CPhill or @mananth, have no that level of flexibility to serve at the adequate level. So, the level of the computer coders must be at least one order (one level) higher. (4) Fourth, the writing, coding and programming discipline must be changed. For now, the AI codes for solving school Math problems that I saw so far, do not show a tendency to make checks of their solutions. They do not think that is is MANDATORY. It is a crude error, which must be fixed. (5) Next. As the life has taught me, many of incoming problems are mathematically non-sensical. So, the input flow is quite dirty, and therefore, idiotic incoming problems should be filtered out.

OK. So, these are conclusions and directions from my overview that should be taken into account by developers of AI,
if they want to have a perfect product in 20 years.

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

OK and very good.

There is one more issue, which I hesitate to pronounce.

It is well known fact that the peak in Math education in the world (in all industrially
developed countries) was from the middle of 1950s to, roughly say, 1970.
After that peak, the recession of the level of Math education was observed worldwide.

So and therefore, the specialists who are able to communicate with young students,
to speak with them, to teach them Math, to inspire them and who got their Math education at that time,
have a special value. People of this generation are quite old and will go out soon -
so and therefore, it is of great importance, great necessity and high urgency to use the experience
of those of them, who still have a potential to share their knowledge, skills, style, tone, spirit and abilities.

When these people will go out totally, it will be too late.

Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
Lauren is planning to raise her batting average. When she set up her weekly practice schedule,
her batting average was 250. Lauren intends to increase her batting average by 5 each week.
Find a pattern and write a formula that will give her expected batting average after n weeks.
a1 =
d=
Using the formula, how many weeks will Lauren need to practice if she expects to earn a batting average of 300?
~~~~~~~~~~~~~~~~~~~~~~

In his post, @mananth produced wrong answer   (21 weeks).

The correct answer to the posed question is    = = 10 weeks,

based on common sense and without blowing the theorizing bubble,   but not 21 weeks.

/\/\/\/\/\/\/\/\/\/\/\/\/

@mananth, theorizing is good, but at the end you should
if it withstands the critics of the common sense.

Without making check, NOTHING prevents you of making errors.

Question 1146477: Jenny puts aside $20 at the end of each month for 3 years. How much will she have then of the investment earns 8.2% p.a., paid monthly?
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
Jenny puts aside $20 at the end of each month for 3 years. How much will she have then of the investment earns
8.2% p.a., paid monthly?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        This my post is written in opposite to the solution by @mananth in his post.

        This Finance problem assumes a precise solution, correct to a single cent.

Use a standard formula for a Future value of an ordinary annuity FV = = = 813.16 dollars. ANSWER

Solved.   In opposite to the answer in the post by @mananth, my answer is precise to one single cent.

To get a precise solution, use MS Excel spreadsheets or Google spreadsheet.
Write the formula in any text editor with the numbers and then copy-paste it into a spreadsheet cell " as is ".

You will get a precise answer with the precision, which is usually enough for
normal/regular financial calculations, without intermediate rounding.

Another, alternative way, is to use specialized financial online calculators.
Many of them, often free of charge and many-times repeatedly tested, can be found in the Internet
using appropriate key words for search, for example "online calculator future value of an annuity".

Below are the links to some popular web-sites with reliable online calculators

https://www.calculatorsoup.com/calculators/financial/future-value-annuity-calculator.php

https://www.calculator.net/annuity-calculator.html

https://www.omnicalculator.com/finance/annuity-future-value

In this concrete calculations, I used MS Excel in my computer.

Happy calculations with reliable tools !

Question 729526: Hi how would you write the equation for the following question?
A 7kW, 240V generator is installed in a photovoltaic system designed with two days of battery storage at 50% depth of discharge(the load uses 50% of the battery storage over 2 days) and the system is sized for a charging rate of C/10.(C/10 = 100% charged over 10 hours C/20 would be 100% charged over 20 hours, etc) Approximately how much fuel will be required per day when the generator is the only power source? The generator burns about 1.6 gallons per hour.
I know the answer is 4 gal per hr.
I would like to see how to write the equation then solve it.
Thank You for your help
Bob
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.

Chop it into small pieces and throw all this garbage in the trash can.
This nonsense does not deserve anything else.

Question 732177: After touchdown, a landing aircraft travels 200 feet the first second, 160 feet the next second, 121 feet the third second, and so on. How many feet will the plane travel in 10 seconds?
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
After touchdown, a landing aircraft travels 200 feet the first second, 160 feet the next second,
121 feet the third second, and so on. How many feet will the plane travel in 10 seconds?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

As it is worded in the post, the "problem" makes no sense.

The words "and so on" in this context have no any reasonable mathematical meaning..

Question 733276: 10 = 1/5 a + 2 solve the equation for a?
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
10 = 1/5 a + 2 solve the equation for a?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Step 1. Subtract 2 from both sides. You will get 10-2 = 1/5 a, or 8 = 1/5 a. Step 2. Multiply both sides of the last equation by 5. You will get 8*5 = a, or a = 40. So, a = 40 is the solution and your ANSWER. CHECK. With a = 40, right side of the given equation is 1/5 40 + 2 = 8 + 2 = 10. So, right side is equal to the left side. It means that the equation is solved correctly.

My congrats !

The answer by @lynnlo is incorrect.

Question 740508: Determine the common ratio of a geometric series that has these partial sums: S4= -3.5, S5= -3.75, S6= -3.875.
I am having a lot of problems trying to find the common ratio. This is a multiple question and none of the answers fit.
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
Determine the common ratio of a geometric series that has these partial sums: S4= -3.5, S5= -3.75, S6= -3.875.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Standard designations mean that S4 is the sum of the first 4 terms of this GP; S5 is the sum of the first 5 terms of this GP; S6 is the sum of the first 6 terms of this GP. Therefore, the 5th terms is S5 - S4 = -3.75 - (-3.5) = -3.75 + 3.5 = -0.25; the 6th terms is S6 - S5 = -3.875 - (-3.75) = -3.875 + 3.75 = -0.125. Now we can easily determine the common ratio of this GP. It is the ratio of the 6th term to the 5th term = 0.5. ANSWER

Solved.

Question 1210418: If (3-x)+(6)+(7-5x) is a geometric series,find two possible values for
a) x
b)the common ratio
c)the sum of the Gp
pls show workings
Found 2 solutions by greenestamps, Edwin McCravy:
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

Another tutor has shown one common method for finding the two possible values of x that produce a geometric sequence, and he has then proceeded to answer all the parts of the question.

Here is an alternative method for finding the two possible values of x and hence of the common ratio. (See his response for the remainder of the problem.)

A very useful method for solving many problems involving geometric sequences is using the fact that if A, B, and C are successive terms of the sequence, then

So in this problem,

or
or

Answer by Edwin McCravy(20077)   (Show Source):
You can put this solution on YOUR website!

If (3-x)+(6)+(7-5x) is a geometric series,find two possible values for a) x b)the common ratio c)the sum of the Gp pls show workings Let the common ratio be r Then Solve the first equation for r: Substitute the right side of that for r in the second equation: 5x+3=0; x-5=0 5x=-3; x=5 x=-3/5 Substitute x=-3/5 in Multiply top and bottom by 5 Substitute x=5 in a)x = -3/5 or 5 b)the common ratio = r = 5/3 or r = -3 c)the sum of the GP: if x = -3/5 if x = 5 ditto ditto ditto Sum of GP = 98/5 or sum of GP = -14 Edwin

Question 488218: The sum of all terms of an infinite geometric progression is 12, and each term is three times the sum of all terms that follow it. What is the first term of the sequence?
Please help. I'm not sure if I'm following it correctly but here's what I've got so far:
Formula: S(infinity)=a1/1-r
where,
S(infinity)=12
a1=3(a2+a3+a4+..an)
And that's about it. I don't know what to do afterwards. :(
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
The sum of all terms of an infinite geometric progression is 12, and each term is three times the sum
of all terms that follow it. What is the first term of the sequence?
~~~~~~~~~~~~~~~~~~~~~~~~~

        Nice problem to wake up your mind.
        I will solve it in a few lines, as simple as I can and as short as it should be.

First equation for the infinite sum is = 12. (1) (I will use standard designations, so I even will not explain them). The first term is three time the sum of all that follow. The terms that follow form geometric progressing starting from with the same common ratio 'r'. So, we write = . (2) We can replace here with . It will give us = . (3) Reduce common factor in both sides. You will get 1 = , 1 - r = 3r ---> 1 = 3r + r, ---> 1 = 4r ---> r = 1/4 = 0.25. Now from equation (1) = 12*(1-r) = = = 3*3 = 9. ANSWER. The first term is = 9, the common ratio is r = 1/4 = 0.25.

Solved completely in 15 lines, including the answer.

Compare it with 58 lines explanations by tutor @Theo.

Question 552039: If a ball rebounds three-fifths as far as it falls, how far will it (vertically) travel before coming to rest if dropped 14 feet?
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
If a ball rebounds three-fifths as far as it falls, how far will it (vertically) travel before coming to rest
if dropped 14 feet?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @Theo is incorrect.
        I came to bring a correct solution.

First rebound is = = ft. The heights of rebounds that follow make geometric progression with the common ratio of r = 3/5. The infinite sum of all the rebound heights series is = = = = 21. This value, 21 ft, we should double, counting each fall-rebund, and add the initial height of 14 ft. So, the final answer for the total travel is 14 + 2*21 = 14 + 42 = 56 ft.

Solved correctly.

Question 1210386: 1, 2, 0, 3, -1, 4, -2
Please enter integer sequence
Found 3 solutions by AnlytcPhil, mccravyedwin, Edwin McCravy:
Answer by AnlytcPhil(1810)   (Show Source):
You can put this solution on YOUR website!

PLEASE NOTICE THAT I HAVE CORRECTED BOTH MY SOLUTIONS ABOVE!

Edwin

Answer by mccravyedwin(421)   (Show Source):
You can put this solution on YOUR website!

Here is another general formula: (You can simplify that if you like, but about all you can do is factor out . How I got the 2nd solution was to find the average of the two solutions for even and odd terms in my first answer above to be . Then I found the common distance from the average for each solution to be . Then I knew that this common distance gets added to the average when is +1, and this common distance gets subtracted from the average when is -1. Edwin

Answer by Edwin McCravy(20077)   (Show Source):
You can put this solution on YOUR website!

simplest formula: Edwin

Question 1210385: 1, 2, 4, 8, 16, 32
Please enter integer sequence
Found 2 solutions by mccravyedwin, Edwin McCravy:
Answer by mccravyedwin(421)   (Show Source):
You can put this solution on YOUR website!

You can write the first one this way You can write the second one this way: Edwin

Answer by Edwin McCravy(20077)   (Show Source):
You can put this solution on YOUR website!

Simplest recursion formula: Simplest general formula Edwin

Question 1169051: The sample of work produced by lathe is 10000.
a. If the machine doubled this production every 20 minutes, how much will it produce
in one hour?
b. How long will it take the machine to produce 8 million?
Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.

        The " solution " by @CPhill is out of logic and out of elementary Arithmetic.

        Here I will give my solution for question (a).

The sample amount is 10,000. An hour is three times 20 minutes. Production for the first 20 minutes 2*10,000 = 20,000. Production for the next 20 minutes 2*20,000 = 40,000. Production for the last 20 minutes 2*40,000 = 80,000. Total production for one hour is 20,000 + 40,000 + 80,000 = 140,000. Answer to question (a)

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Let the initial production of the lathe be $P_0 = 10000$.
**a. Production in one hour:**
One hour has 60 minutes. The machine doubles its production every 20 minutes. So, in one hour, the production will double $60 / 20 = 3$ times.
* After the first 20 minutes: $P_1 = P_0 \times 2 = 10000 \times 2 = 20000$
* After the next 20 minutes (total 40 minutes): $P_2 = P_1 \times 2 = 20000 \times 2 = 40000$
* After the final 20 minutes (total 60 minutes or one hour): $P_3 = P_2 \times 2 = 40000 \times 2 = 80000$
Alternatively, we can use the formula:
$P(t) = P_0 \times 2^{t/d}$
where:
* $P(t)$ is the production after time $t$
* $P_0$ is the initial production
* $t$ is the total time
* $d$ is the doubling time
For one hour (60 minutes):
$P(60) = 10000 \times 2^{60/20}$
$P(60) = 10000 \times 2^3$
$P(60) = 10000 \times 8$
$P(60) = 80000$
So, the machine will produce **80,000** units in one hour.
**b. Time to produce 8 million:**
We want to find the time $t$ when the production $P(t)$ reaches 8,000,000.
$P(t) = 10000 \times 2^{t/20} = 8000000$
Divide both sides by 10000:
$2^{t/20} = \frac{8000000}{10000}$
$2^{t/20} = 800$
To solve for $t$, we can take the logarithm of both sides (using base 2 or natural logarithm):
Using base 2 logarithm:
$\log_2(2^{t/20}) = \log_2(800)$
$\frac{t}{20} = \log_2(800)$
We know that $2^9 = 512$ and $2^{10} = 1024$. So, $\log_2(800)$ is between 9 and 10.
$\log_2(800) = \log_2(8 \times 100) = \log_2(2^3 \times 100) = 3 + \log_2(100)$
Since $2^6 = 64$ and $2^7 = 128$, $\log_2(100)$ is between 6 and 7 (approximately 6.64).
$\log_2(800) \approx 3 + 6.64 = 9.64$
Now, solve for $t$:
$t = 20 \times \log_2(800)$
$t \approx 20 \times 9.64$
$t \approx 192.8$ minutes
Alternatively, using natural logarithm:
$\ln(2^{t/20}) = \ln(800)$
$\frac{t}{20} \ln(2) = \ln(800)$
$t = 20 \times \frac{\ln(800)}{\ln(2)}$
$t \approx 20 \times \frac{6.6846}{0.6931}$
$t \approx 20 \times 9.644$
$t \approx 192.88$ minutes
So, it will take approximately **192.88 minutes** for the machine to produce 8 million units.
To express this in hours and minutes:
$192.88 \text{ minutes} = 3 \text{ hours and } 12.88 \text{ minutes}$
$0.88 \text{ minutes} \times 60 \text{ seconds/minute} \approx 53 \text{ seconds}$
So, it will take approximately 3 hours, 12 minutes, and 53 seconds.
Final Answers:
a. The machine will produce **80,000** units in one hour.
b. It will take approximately **192.88 minutes** (or about 3 hours, 12 minutes, and 53 seconds) for the machine to produce 8 million units.

Question 1210345: THE 2ND AND 7TH TERM OF A G.P ARE 18 AND 4374 RESPECTIVELY. FIND THE
1) COMMON DIFFERENCE
2) FIRST TERM
3) SUM OF THE 4TH AND 8TH TERM
4) SUM OF THE FIRST 10 TERMS
Found 5 solutions by AnlytcPhil, ikleyn, mccravyedwin, Edwin McCravy, josgarithmetic:
Answer by AnlytcPhil(1810)   (Show Source):
You can put this solution on YOUR website!

Ikleyn says: To avoid misunderstanding, here is my correction to the post by Edwin, where he placed his solution for Geometric progression. His last formula in his post should be = = = 6*29524 = 177144. And if you'll notice, that's exactly what I had Parentheses are not necessary. Edwin

Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.

To avoid misunderstanding, here is my correction to the post by Edwin,
where he placed his solution for Geometric progression.

His last formula in his post should be

= = = 6*29524 = 177144.

////////////////////////////////

As I see from the Edwin's post (as @AnlytcPhil), Edwin did not get the meaning of my correction.

The meaning is not in placing parentheses.

The meaning is that I changed ' 10-1 ' in the denominator by ' 3-1 ',
and then changed ' 9 ' in the denominator by ' 2 ' to make the numbers consistent.

Answer by mccravyedwin(421)   (Show Source):
You can put this solution on YOUR website!

Now I'll assume the student meant THE 2ND AND 7TH TERM OF AN A.P ARE 18 AND 4374 RESPECTIVELY. FIND THE 1) COMMON DIFFERENCE 2) FIRST TERM 3) SUM OF THE 4TH AND 8TH TERM 4) SUM OF THE FIRST 10 TERMS I'll use 1) COMMON DIFFERENCE Subtract the first equation term-by-term from the second equation. 2) FIRST TERM 3) SUM OF THE 4TH AND 8TH TERM Using 4) SUM OF THE FIRST 10 TERMS Edwin

Answer by Edwin McCravy(20077)   (Show Source):
You can put this solution on YOUR website!

I'll assume the student meant THE 2ND AND 7TH TERM OF A G.P ARE 18 AND 4374 RESPECTIVELY. FIND THE 1) COMMON RATIO 2) FIRST TERM 3) SUM OF THE 4TH AND 8TH TERM 4) SUM OF THE FIRST 10 TERMS 1) COMMON RATIO Divide the second equation term-by-term by the first equation. 2) FIRST TERM 3) SUM OF THE 4TH AND 8TH TERM Using 4) SUM OF THE FIRST 10 TERMS Edwin

Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
What kind of pattern is this? "A G P ARE..."?
Geometric Progression? Why you ask for "COMMON DIFFERENCE"?

Question 1209826: For a positive integer k, let
S_k = 1 \cdot 1! \cdot 2 + 2 \cdot 2! \cdot 3 + \dots + k \cdot k! \cdot (k + 1).
Find a closed form for S_k.
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
S_k = 1*(1^2) + 2!*(2^2)*3 + . . . + k*k!*(k + 1).
Find a closed form for S_k.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution and the answer in the post by @CPhill both are .

                        Indeed,      let's check for k = 3.

Left side is

            1!*(1^2) + 2!*(2^2)*3 + 3*3!*(3+1) = 1*(1) + 2*(4)*3 + 3*(6)*4 = 1 + 24 + 72 = 97.

Right side, according to @CPhill, is

            3*(3+1)! - 2 = 3*4! - 2 = 3*24 - 2 = 72 - 2 = 70.

But   97 =/= 70.

This is the , which ruins the solution by @CPhill to dust.

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

                Regarding the post by @CPhill . . .

Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.

The artificial intelligence is like a baby now. It is in the experimental stage
of development and can make mistakes and produce nonsense without any embarrassment.

                It has no feeling of shame - it is shameless.

This time, again, it made an error.

Although the @CPhill' solutions are copy-paste Google AI solutions, there is one essential difference.

Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.

All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.

Every time, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.

And the last my comment.

When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.

Without it, their reliability is ZERO and their creadability is ZERO, too.

Question 1209827: Find a closed form for
S_n = 1! \cdot (1^2 + 1) + 2! \cdot (2^2 + 2) + \dots + n! \cdot (n^2 + n).\]
for any integer n \ge 1. Your response should have a factorial.
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
Find a closed form for S_n = 1!*(1^2 + 1) + 2!*(2^2 + 2) + . . . + n!*(n^2 + n)
for any integer n >= 1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution and the answer in the post by @CPhill both are .

                        Indeed,      let's check for n = 3.

Left side is

            1!*(1^2+1) + 2!*(2^2+2) + 3!*(3^2+3) = 1*(1+1) + 2*(2+2) + 6*(9+3) = 1*2 + 2*4 + 6*12 = 2 + 8 + 72 = 82.

Right side, according to @CPhill, is

            (3+2)! - 2 = 5! - 2 = 1*2*3*4*5 - 2 = 120 - 2 = 118.

But   82 =/= 118.

This is the , which ruins the solution by @CPhill to dust.

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

                Regarding the post by @CPhill . . .

Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.

The artificial intelligence is like a baby now. It is in the experimental stage
of development and can make mistakes and produce nonsense without any embarrassment.

                It has no feeling of shame - it is shameless.

This time, again, it made an error.

Although the @CPhill' solutions are copy-paste Google AI solutions, there is one essential difference.

Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.

All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.

Every time, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.

And the last my comment.

When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.

Without it, their reliability is ZERO and their creadability is ZERO, too.

Question 1209805: Let a_1 + a_2 + a_3 + dotsb be an infinite geometric series with positive terms. If a_2 = 10, then find the smallest possible value of
a_1 + a_2 + a_3.
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
Let a_1 + a_2 + a_3 + dots be an infinite geometric series with positive terms.
If a_2 = 10, then find the smallest possible value of a_1 + a_2 + a_3.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        This problem is simple and elementary, and I will show below
        a simple solution without using Calculus and/or derivatives.

The fact that this geometric progression has positive terms tells us that the first term is positive and the common ratio is positive, too. So, the sum can be presented in the form + + = + 10 + 10*r. (1) We can identically transform this expression in the right side of (1) this way + 10 + 10r = ( - 20 + 10r) + 30 = + 30. (2) Now, the part is always greater than or equal to zero, since it is the square of real number. Hence, this expression is minimal if and only if = , (3) when is equal to zero. Square both sides in (3) = 10r, or = r, --> 1 = r^2 --> r = = 1. Hence, the sum (1) is minimal if and only if r = 1. Then the sum (1) is + 10 + 10*1 = 10 + 10 + 10 = 30. At this point, the solution is complete. ANSWER. The sum of geometric progression with positive terms is minimal if and only if the common ratio r is 1. It is the case when all three terms of the progression are equal. For our case, this minimal value of the sum of the first three terms is 30, i.e. thrice its central term.

Solved completely.

----------------------------

As this problem is worded and presented, it considers only three first terms of the geometric progression.

Therefore, in the problem's formulation, there is no any need to consider an infinite progression.

Good style tells us to consider only three-term geometric progression from the very beginning.

Moreover, an infinite geometric progression with r= 1 diverges and its sum does not exist (is infinity).

Question 1179819: f. Find the present values of the following annuities
i. RM6,000 every year for 8 years at 12% compounded annually
ii. RM800 every month for 2 years 5 months at 5% compounded monthly

Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
f. Find the present values of the following annuities
i. RM6,000 every year for 8 years at 12% compounded annually
ii. RM800 every month for 2 years 5 months at 5% compounded monthly
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I checked calculations by @CPhill.

For calculations, I used MS Excel in my computer. This software is commonly considered
as a standard tool, which provides the necessary precision for such calculations.

For (i), I got the same value, so this part is correct.

For (ii), I got different value of RM 21810.41.

In this case, I used formula

        PV = .

I did not perform intermediate rounding: it is PROXIBITED in such calculations
and makes big influence, leading to incorrect answer.

I copy/pasted this my formula into Excel spreadsheet as is and got the answer in the next instance.

The difference in our predictions is   21810.41 - 21571.39 = 239 RM.

Such a great discrepancy in Finance calculations is not allowable and is not acceptable.

Overviewing calculations by @CPhill in many other his posts, I concluded,
that he is irresponsible in his calculations and does not care about their precision.

Question 1210233: Show that the sum of n terms of the progression
log(x), log(x^2), log(x^3), log(x^4) , ..., log(x^n) is (n*(n+1)/2)*log x.
Found 3 solutions by mccravyedwin, ikleyn, Edwin McCravy:
Answer by mccravyedwin(421)   (Show Source):
You can put this solution on YOUR website!

Ikleyn's solution is correct as she interpreted it. Sometimes English is not the first language of the student, and the way they translate things into English is not always the same way we express things in the US. Edwin

Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
Show that the sum of the end term of the progression Log x, Log x^2, Log x^3, log x^4 = n (n+1/2) Log x
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        Wording and writing in the post are incorrect, so I edited it to make sense from nonsense.

        My edited formulation is as follows:

                Show that the sum of n terms of the progression log(x), log(x^2), log(x^3), log(x^4) , . . , log(x^n) is (n*(n+1)/2)*log x.

        Below is my solution for this edited formulation.

In this problem, x > 0. Let a = log(x). Then = 2a, = 3a, . . . and so on . . . till = n*a. Therefore, our progression takes the form a + 2a + 3a + . . . + n*a = (1 + 2 + 3 + . . . + n)*a. The sum in parentheses in the right side is well known sum of the arithmetic progression, and it is equal to . Thus + + + . . . + = . It is the final answer, and the proof is complete.

Solved.

Answer by Edwin McCravy(20077)   (Show Source):
You can put this solution on YOUR website!

I did not understand the problem, so I deleted what I thought it meant. Ikleyn's solution is correct. Notice that I changed the wording of the original problem to her edited wording. I think we tutors should change the wording of the original problem whenever it is not easily understandable as written. Edwin

Question 1168274: what is the first five terms and 50th term of this sequence.
An=2a n-1 + 5 and a1=3
Found 3 solutions by Edwin McCravy, greenestamps, ikleyn:
Answer by Edwin McCravy(20077)   (Show Source):
You can put this solution on YOUR website!

I wrote a program in LibertyBasic using the given recursion formula and greenestamps' general formula. Here is the LibertyBasic progrem: for n=1 to 50 if n=1 then a=3: goto 1 a=2*a+5 1 print n, a, 2^(n+2)-5 next They are identical, as you can see from the complete output below. However, the 50th term does not agree with greenestamps' 50th term. He made a slight calculator error and got the 51st term by calculating 2^53-5 instead of 2^52-5. 2a(n-1)-5 a(1)=3 a(n)=2ⁿ⁺²-5 1 3 3 2 11 11 3 27 27 4 59 59 5 123 123 6 251 251 7 507 507 8 1019 1019 9 2043 2043 10 4091 4091 11 8187 8187 12 16379 16379 13 32763 32763 14 65531 65531 15 131067 131067 16 262139 262139 17 524283 524283 18 1048571 1048571 19 2097147 2097147 20 4194299 4194299 21 8388603 8388603 22 16777211 16777211 23 33554427 33554427 24 67108859 67108859 25 134217723 134217723 26 268435451 268435451 27 536870907 536870907 28 1073741819 1073741819 29 2147483643 2147483643 30 4294967291 4294967291 31 8589934587 8589934587 32 17179869179 17179869179 33 34359738363 34359738363 34 68719476731 68719476731 35 137438953467 137438953467 36 274877906939 274877906939 37 549755813883 549755813883 38 1099511627771 1099511627771 39 2199023255547 2199023255547 40 4398046511099 4398046511099 41 8796093022203 8796093022203 42 17592186044411 17592186044411 43 35184372088827 35184372088827 44 70368744177659 70368744177659 45 140737488355323 140737488355323 46 281474976710651 281474976710651 47 562949953421307 562949953421307 48 1125899906842619 1125899906842619 49 2251799813685243 2251799813685243 50 4503599627370491 4503599627370491 Edwin

Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

The recursive formula says we get the next term in the sequence by doubling the current term and adding 5. Starting with 3 as the first term...

a(1) = 3
a(2) = 3(2)+5 = 11
a(3) = 11(2)+5 = 27
a(4) = 27(2)+5 = 59
a(5) = 59(2)+5 = 123

It would be very tedious to find the 50th term by continuing to use the recursive formula, so to find the 50th term we would like to find an explicit formula for the n-th term.

To do that, we could try the method of finite differences. Here are the first few terms of the sequence and the first and second differences:
3 11 27 59 123 8 16 32 64 8 16 32

That pattern of differences tells us that the formula for the n-th term is not a polynomial but instead is based on powers of 2.

Playing with the first few numbers in the sequence, we see that adding 5 to each term gives us a sequence of powers of 2:

8, 16, 32, 64, 128, ...

So...
a(1) is 8-5 = 2^3-5
a(2) is 16-5 = 2^4-5
a(3) is 32-5 = 2^5-5
...

And we see the pattern for the n-th term is

a(n) = 2^(n+2)-5

So the 50th term is

2^52-5 = 9007199254740987

My TI-84 calculator won't calculate that number; and I tried using excel but it fell one digit short of being able to calculate the exact number. I used an online program (pari) to find the exact answer.

Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.

In this problem, you are given a recurrent formula to calculate next term of the sequence using its previous step. You also are given the first term. So, to get the second term, simply substitute first term a1 = 3 into the formula a2 = 2*3 + 5 = 6 + 5 = 11. To get the third term, substitute the second term a2 = 11 into the formula a3 = 2*11 + 5 =22 + 5 = 27. Continue doing this way until you will get a5.

Question 1210209: If a,b,c are in Harmonic Progression, show that 1/a + 1/(b+c) , 1/b + 1/(c+a) , 1/c + 1/(a+b) are also in Harmonic Progression
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Since $a, b, c$ are in Harmonic Progression (HP), their reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in Arithmetic Progression (AP).
This means that the difference between consecutive terms is constant:
$\frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b}$
$\frac{a-b}{ab} = \frac{b-c}{bc}$
$bc(a-b) = ab(b-c)$
Dividing by $abc$ (assuming $a, b, c \neq 0$), we get:
$\frac{a-b}{a} = \frac{b-c}{c}$
$c(a-b) = a(b-c)$
$ca - cb = ab - ac$
$2ac = ab + bc$
Now, let's consider the terms $\frac{1}{a} + \frac{1}{b+c}$, $\frac{1}{b} + \frac{1}{c+a}$, $\frac{1}{c} + \frac{1}{a+b}$.
For these terms to be in Harmonic Progression, their reciprocals must be in Arithmetic Progression. Let's find the reciprocals:
$x = \frac{1}{\frac{1}{a} + \frac{1}{b+c}} = \frac{a(b+c)}{b+c+a}$
$y = \frac{1}{\frac{1}{b} + \frac{1}{c+a}} = \frac{b(c+a)}{c+a+b}$
$z = \frac{1}{\frac{1}{c} + \frac{1}{a+b}} = \frac{c(a+b)}{a+b+c}$
For $x, y, z$ to be in AP, we need $y - x = z - y$, or $2y = x + z$.
$2 \left( \frac{b(c+a)}{a+b+c} \right) = \frac{a(b+c)}{a+b+c} + \frac{c(a+b)}{a+b+c}$
Since the denominators are the same, we can equate the numerators:
$2b(c+a) = a(b+c) + c(a+b)$
$2bc + 2ab = ab + ac + ca + cb$
$2bc + 2ab = ab + 2ac + cb$
$bc + ab = 2ac$
This is the same condition we derived from the fact that $a, b, c$ are in HP.
Therefore, the reciprocals of $\frac{1}{a} + \frac{1}{b+c}$, $\frac{1}{b} + \frac{1}{c+a}$, $\frac{1}{c} + \frac{1}{a+b}$ are in Arithmetic Progression, which means that $\frac{1}{a} + \frac{1}{b+c}$, $\frac{1}{b} + \frac{1}{c+a}$, $\frac{1}{c} + \frac{1}{a+b}$ are in Harmonic Progression.
Final Answer: The final answer is $\boxed{\frac{1}{a} + \frac{1}{b+c} , \frac{1}{b} + \frac{1}{c+a} , \frac{1}{c} + \frac{1}{a+b} \text{ are in Harmonic Progression}}$

Question 1210195: If a,b,c are in Harmonic Progression,prove that 1/a + 1/(b+c), 1/b + 1(c+a), 1/c + 1(a+b) are also in Harmonic Progression
Answer by mccravyedwin(421)   (Show Source):
You can put this solution on YOUR website!

If a,b,c are in Harmonic Progression, That means that 1/a, 1/b, 1/c are in arithmetic progression, i.e. We are to show that (1) All terms are of the form Multiply top and bottom by x(y+z) So (1) becomes (2) (2) will be true if and only if which will be true if and only if which will be true if and only if which will be true if and only if Substitute which will be true if and only if Which will be true if and only if And since this is true, the problem is proved. Edwin

Question 1168320: Suppose the yearly inflation rate from 2014 to 2020 is 15%, the table that costs $800 at the start of 2014 costs $920 at the start of 2020, and so on. What equation represents the cost of the table from the year 2014 to 2020?
Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
Suppose the yearly inflation rate from 2014 to 2020 is 15%, the table that costs $800 at the start of 2014
costs $920 at the start of 2020, and so on. What equation represents the cost of the table from the year 2014 to 2020?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Cost(2020) = Cost(2014)*(1+0.15) = 800*1.15. ANSWER

Solved.

Question 1209977: For a positive integer n, let f(n) denote the integer that is closest to
. Find the integer m so that
.

Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20077)   (Show Source):
You can put this solution on YOUR website!

For a positive integer n, let f(n) denote the integer that is closest to . Find the integer m so that . So terms 1 through 5 are all 1's. That's 5-1+1 = 5 terms of 1's We have 5 terms so far. And so far the sum is 5x1 = 5. So terms 6 through 39 are all 2's. That's 39-6+1=34 terms of 2's. We have 5+34=39 terms so far. And so far the sum is 5 + 34x2 = 5 + 68 = 73 So terms from 39 on are all 3's. So we only need to know how many more terms of 3's we need beyond the 39 terms to raise the sum of 73 to 100. We need to raise the sum by 100-73=27 which will require 27/3=9 more terms of 3's. So the total number of terms will be 39+9 = 48. So m = 48 Edwin

Answer by ikleyn(53749)   (Show Source):
You can put this solution on YOUR website!
.
For a positive integer n, let f(n) denote the integer that is closest to sqrt[4]{n}.
Find the integer m so that sum_{n = 1}^m f(n) = 100.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        I solved this problem using MS Excel.
        My calculations are shown in the table below.

First column of the table in the counter of natural numbers n = 1, 2, 3, . . . Second column is the values of , rounded to the closest integer number. Third column is the sum S(n) of the first n integer numbers of the second column. The table shows that the integer 'm' such that the sum S(m) is precisely equal to 100 is 48. n S(n) ----------------------------------------------- 1 1 1 2 1 2 3 1 3 4 1 4 5 1 5 6 2 7 7 2 9 8 2 11 9 2 13 10 2 15 11 2 17 12 2 19 13 2 21 14 2 23 15 2 25 16 2 27 17 2 29 18 2 31 19 2 33 20 2 35 21 2 37 22 2 39 23 2 41 24 2 43 25 2 45 26 2 47 27 2 49 28 2 51 29 2 53 30 2 55 31 2 57 32 2 59 33 2 61 34 2 63 35 2 65 36 2 67 37 2 69 38 2 71 39 2 73 40 3 76 41 3 79 42 3 82 43 3 85 44 3 88 45 3 91 46 3 94 47 3 97 48 3 100 <<<---=== So, the ANSWER to the problem's question is m = 48. Having this table, one can construct a wording solution, retelling this my solution in wording form without using this table, but I prefer direct arguments.

Solved.