Tutors Answer Your Questions about Sequences-and-series (FREE)
Question 1193591: Find the sum of this series that is not arithmetic or geometric,
1+2+4+5+7+8....+95+97+98
can someone help, a few of us have been trying to figure this out, our teacher said its solvable but I don't even know where to start, thankyou
Click here to see answer by math_tutor2020(3817)  |
Question 1193591: Find the sum of this series that is not arithmetic or geometric,
1+2+4+5+7+8....+95+97+98
can someone help, a few of us have been trying to figure this out, our teacher said its solvable but I don't even know where to start, thankyou
Click here to see answer by ikleyn(52787)  |
Question 1193591: Find the sum of this series that is not arithmetic or geometric,
1+2+4+5+7+8....+95+97+98
can someone help, a few of us have been trying to figure this out, our teacher said its solvable but I don't even know where to start, thankyou
Click here to see answer by greenestamps(13200)  |
Question 1194395: I starting to learn about mathematical induction, and I'm not sure if my answers are correct. Please help.
Here is the question, my answers are inside brackets, the rest is provided by the exercise. Thank you.
Complete the following proof by mathematical induction that, for all n in W such that n >= 2, 3 ^ n > 2 ^ (n + 1)
We proceed by mathematical induction and begin by establishing the base of the induction.
3^ 2 = [9] > [8] =2^ 2+1
We see that 3 ^ n > 2 ^ (n + 1) when n=2.
Moving on to the induction step, we suppose that m in W is such that m >= 2
and as our induction hypothesis, we take the assumption that 3^m > 2 ^ (m + 1)
Noting that 3m+is the product of one more 3 than 3 ^m+1 and that 2 ^ (m + 2) is the product of one more 2 than 2 ^ (m + 1)
We calculate as follows:
2 ^ (m + 1) + 1 = [ 2 * 2 ^ (m + 1) ]
(choose from; 2+2^(m+1), 2.2^(m+1), 2^(m+1) +1)
= [ 2^ m+1 +2^ m+1 ]
(choose from; 2^m*2^m, 2^m+2^m, 2^(m+1)+2^(m+1))
< 3^m +2^m+1 by the induction hypothesis and additive monotonicity
< [ 3^m + 2 ^ m + 2^m ] by the induction hypothesis and additive monotonicity
(choose from; 3^m + 2^m + 2^m, 3^m +3^m, 3^m + 3^(m+1) )
< [ 3^m + 3^m + 3^m ] by order in W since 3^ m is in N
(choose from; 3^m *3^m, 3m +3^(m+1), 3^m + 3^m + 3^m )
= [3*3^m ]
(choose from, 3*3^m, 3^m*3^m, 3+3^m )
=3^ m+1
We have shown that 3 ^ 2 > 2 ^ (2 + 1) and that, for all m in W such that m >= 2, if * 3 ^ m > 2 ^ (m + 1)
then 3 ^ (m + 1) > 2 ^ (m + 1) + 1
We may therefore conclude by mathematical induction that, for all n in W such that n >= 2; 3 ^ n > 2 ^ (n + 1)
Click here to see answer by math_helper(2461)  |
Question 1194395: I starting to learn about mathematical induction, and I'm not sure if my answers are correct. Please help.
Here is the question, my answers are inside brackets, the rest is provided by the exercise. Thank you.
Complete the following proof by mathematical induction that, for all n in W such that n >= 2, 3 ^ n > 2 ^ (n + 1)
We proceed by mathematical induction and begin by establishing the base of the induction.
3^ 2 = [9] > [8] =2^ 2+1
We see that 3 ^ n > 2 ^ (n + 1) when n=2.
Moving on to the induction step, we suppose that m in W is such that m >= 2
and as our induction hypothesis, we take the assumption that 3^m > 2 ^ (m + 1)
Noting that 3m+is the product of one more 3 than 3 ^m+1 and that 2 ^ (m + 2) is the product of one more 2 than 2 ^ (m + 1)
We calculate as follows:
2 ^ (m + 1) + 1 = [ 2 * 2 ^ (m + 1) ]
(choose from; 2+2^(m+1), 2.2^(m+1), 2^(m+1) +1)
= [ 2^ m+1 +2^ m+1 ]
(choose from; 2^m*2^m, 2^m+2^m, 2^(m+1)+2^(m+1))
< 3^m +2^m+1 by the induction hypothesis and additive monotonicity
< [ 3^m + 2 ^ m + 2^m ] by the induction hypothesis and additive monotonicity
(choose from; 3^m + 2^m + 2^m, 3^m +3^m, 3^m + 3^(m+1) )
< [ 3^m + 3^m + 3^m ] by order in W since 3^ m is in N
(choose from; 3^m *3^m, 3m +3^(m+1), 3^m + 3^m + 3^m )
= [3*3^m ]
(choose from, 3*3^m, 3^m*3^m, 3+3^m )
=3^ m+1
We have shown that 3 ^ 2 > 2 ^ (2 + 1) and that, for all m in W such that m >= 2, if * 3 ^ m > 2 ^ (m + 1)
then 3 ^ (m + 1) > 2 ^ (m + 1) + 1
We may therefore conclude by mathematical induction that, for all n in W such that n >= 2; 3 ^ n > 2 ^ (n + 1)
Click here to see answer by ikleyn(52787) |
Question 1194555: You make payments of $150 at the end of each month into a savings account that earns 3% annual interest. How long will it take for the balance to reach $6000? Round your answer to the nearest whole number of months.
Click here to see answer by ikleyn(52787) |
Question 1194564: Suppose you go to a company that pays 0.01 for the first day, 0.02 for the second day, 0.04 for the third day and so on. If the daily wage keeps doubling, what will your total income be for working 29 days ?
Click here to see answer by ikleyn(52787) |
Question 1194564: Suppose you go to a company that pays 0.01 for the first day, 0.02 for the second day, 0.04 for the third day and so on. If the daily wage keeps doubling, what will your total income be for working 29 days ?
Click here to see answer by Boreal(15235)  |
Question 1194628: You have a saving $350,000 in a retirement account that earns 4.8% annual interest. You want the savings to last 20 years. How much money can you withdraw each month?
Suppose that you instead want to withdraw $3000 per month. How long will your savings last? Round you answer up to the next whole number of months.
Click here to see answer by Theo(13342)  |
Question 1194741: You want to be able to withdraw $45,000 each year for 20 years. Your account earns 10% interest.
a) How much do you need in your account at the beginning?
$
b) How much total money will you pull out of the account?
$
c) How much of that money is interest?
$
Click here to see answer by Boreal(15235) |
Question 1194625: You withdraw $1500 from a bank account each month that earns 3% annual interest. How much money do you need to start with in the account so that it will last 20 years? Round your answer to 2 decimal places
Click here to see answer by Theo(13342) |
Question 1195429: Number Theory
2.(2017 Putnam)Evaluate the sum
\[
\sum_{k=0}^{\infty} \left( 3\cdot \frac{ln(4k+2)}{4k+2}-\frac{ln(4k+3)}{4k+3}-\frac{ln(4k+4)}{4k+4}-\frac{ln(4k+5)}{4k+5} \right)\] \hfill \break
\[=3\cdot \frac{ln2}{2}-\frac{ln3}{3}-\frac{ln4}{4}-\frac{ln5}{5}-\frac{ln6}{6}-\frac{ln7}{7}-\frac{ln8}{8}-\frac{ln93}{9}+3\cdot\frac{ln10}{10}-...
\]
\textbf{(As usual, ln $x$ denotes the natural logarithm of $x$.}
to translate to an image, go to https://artofproblemsolving.com/texer/nrqjrtmo
Click here to see answer by ikleyn(52787) |
Question 1195428: Number Theory( I don't know where to put this, but since this has the symbol sigma in it, I put it here)
1. (2015 Putnam) For each positive integer k, let A(k) be the number of odd divisors of k in the interval [1, \large \sqrt{2k}]]. Evaluate
\[ \sum_{k=1}^{\infty} (-1)^{k-1} \frac{A(k)}{k} \
Note: this is typed in latex, you could view it more properly here:
https://artofproblemsolving.com/texer/zrydkqmo
Click here to see answer by ikleyn(52787) |
Question 1195444: Hi Team, Kindly help me to find the next number of the sequence.
339834,945298,227146,267298,476197,150978,826467,750087,??
722556,129222,237263,593553,??
Thanks,
Sam
samareshsam18@gmail.com
Click here to see answer by ikleyn(52787) |
Question 1195483: Dear teaches, my math teacher gave me the following sequence: 1234321, 26121262, 1241212121212, ?
The task is to solve the next term (the one with an ?)
He said that you don't need any advanced math to solve it, but only pattern recognition skill. So it doesn't necessarily must have a formula, just a logical pattern (for example, between digits like the sequnce 9328, 1627, 1406 where the pattern here is that the product of the two first digits of a term are the last two digits of the next term (9*3=27, 1*6=06) and the product of the last two digits of a term are in the first two digits of the next term (2*8=16, 2*7=14).
I've tried very hard but still couldn't solved it. What I've seen, is the next relation between the 1st and 2nd term; 1*2=2, 2*3=6, 3*4=12, 4*3=12, 3*2=6, 2*1=2. So it seems that the 2nd term is a compound of numbers (2,6,12,12,6,2) made of from the product of the digits or numbers of the first term (1,2,3,4,3,2,1). But then from the 2nd to the 3rd term seems like the pattern changes a little (maybe multiplying from different direction, right to left).
Maybe I'm looking in the wrong place so don't let my analysis focus on you, but more like an observation.
And remember, all what you need is a bright idea.
I hope I made it clear, and thanks in advance for your attention. It is really impotant for me to understand this sequence.
Click here to see answer by greenestamps(13200) |
Question 1195753: Identify the number pattern and fill in the missing numbers.
A-5,3,9,2,13
B-24,23,25,26,26
C-14,17,13,15,12
D-74,70,77,75,80
E-32,36,31,38,30
F-1,10,3,12,5
G-56,52,54,55,52
H-17,21,13,23,9
I-90,10,85,15,80
J-67,70,64,68,61
Click here to see answer by greenestamps(13200) |
Question 1195915: First number sequence:
75,15,25,5,15, ...
The correct next number is 3
I don't get why?
Second sequence:
183 305 527 749 961 ...
The correct next number is 293.
I also can't solve it.
Please help to understand how they both work?
Click here to see answer by greenestamps(13200) |
Question 1195958: I have a sequence (as a riddle from a friend), but I don't know the pattern(s)/rule(s): 8, 12, 18, 24, 38, 60, 98, 150, 240, 380, 614, Can you help me? Regards, Michael
Click here to see answer by ikleyn(52787) |
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Older solutions: 1..45, 46..90, 91..135, 136..180, 181..225, 226..270, 271..315, 316..360, 361..405, 406..450, 451..495, 496..540, 541..585, 586..630, 631..675, 676..720, 721..765, 766..810, 811..855, 856..900, 901..945, 946..990, 991..1035, 1036..1080, 1081..1125, 1126..1170, 1171..1215, 1216..1260, 1261..1305, 1306..1350, 1351..1395, 1396..1440, 1441..1485, 1486..1530, 1531..1575, 1576..1620, 1621..1665, 1666..1710, 1711..1755, 1756..1800, 1801..1845, 1846..1890, 1891..1935, 1936..1980, 1981..2025, 2026..2070, 2071..2115, 2116..2160, 2161..2205, 2206..2250, 2251..2295, 2296..2340, 2341..2385, 2386..2430, 2431..2475, 2476..2520, 2521..2565, 2566..2610, 2611..2655, 2656..2700, 2701..2745, 2746..2790, 2791..2835, 2836..2880, 2881..2925, 2926..2970, 2971..3015, 3016..3060, 3061..3105, 3106..3150, 3151..3195, 3196..3240, 3241..3285, 3286..3330, 3331..3375, 3376..3420, 3421..3465, 3466..3510, 3511..3555, 3556..3600, 3601..3645, 3646..3690, 3691..3735, 3736..3780, 3781..3825, 3826..3870, 3871..3915, 3916..3960, 3961..4005, 4006..4050, 4051..4095, 4096..4140, 4141..4185, 4186..4230, 4231..4275, 4276..4320, 4321..4365, 4366..4410, 4411..4455, 4456..4500, 4501..4545, 4546..4590, 4591..4635, 4636..4680, 4681..4725, 4726..4770, 4771..4815, 4816..4860, 4861..4905, 4906..4950, 4951..4995, 4996..5040, 5041..5085, 5086..5130, 5131..5175, 5176..5220, 5221..5265, 5266..5310, 5311..5355, 5356..5400, 5401..5445, 5446..5490, 5491..5535, 5536..5580, 5581..5625, 5626..5670, 5671..5715, 5716..5760, 5761..5805, 5806..5850, 5851..5895, 5896..5940, 5941..5985, 5986..6030, 6031..6075, 6076..6120, 6121..6165, 6166..6210, 6211..6255, 6256..6300, 6301..6345, 6346..6390, 6391..6435, 6436..6480, 6481..6525, 6526..6570, 6571..6615, 6616..6660, 6661..6705, 6706..6750, 6751..6795, 6796..6840, 6841..6885, 6886..6930, 6931..6975, 6976..7020, 7021..7065, 7066..7110, 7111..7155, 7156..7200, 7201..7245, 7246..7290, 7291..7335, 7336..7380, 7381..7425, 7426..7470, 7471..7515, 7516..7560, 7561..7605, 7606..7650, 7651..7695, 7696..7740, 7741..7785, 7786..7830, 7831..7875, 7876..7920, 7921..7965, 7966..8010, 8011..8055, 8056..8100, 8101..8145, 8146..8190, 8191..8235, 8236..8280, 8281..8325, 8326..8370, 8371..8415, 8416..8460, 8461..8505, 8506..8550, 8551..8595, 8596..8640, 8641..8685, 8686..8730, 8731..8775, 8776..8820, 8821..8865, 8866..8910, 8911..8955, 8956..9000, 9001..9045, 9046..9090, 9091..9135, 9136..9180, 9181..9225, 9226..9270, 9271..9315, 9316..9360, 9361..9405, 9406..9450, 9451..9495, 9496..9540, 9541..9585, 9586..9630, 9631..9675, 9676..9720, 9721..9765, 9766..9810, 9811..9855, 9856..9900, 9901..9945, 9946..9990, 9991..10035, 10036..10080, 10081..10125, 10126..10170, 10171..10215, 10216..10260, 10261..10305, 10306..10350, 10351..10395, 10396..10440, 10441..10485, 10486..10530, 10531..10575, 10576..10620, 10621..10665, 10666..10710, 10711..10755, 10756..10800, 10801..10845, 10846..10890, 10891..10935, 10936..10980, 10981..11025, 11026..11070, 11071..11115, 11116..11160, 11161..11205, 11206..11250, 11251..11295, 11296..11340, 11341..11385, 11386..11430, 11431..11475, 11476..11520, 11521..11565, 11566..11610, 11611..11655, 11656..11700, 11701..11745, 11746..11790
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