SOLUTION: 1 + 4 + 7 + 10 + ... + x = 287 find x

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Question 1209569: 1 + 4 + 7 + 10 + ... + x = 287
find x
Found 4 solutions by josgarithmetic, ikleyn, math_tutor2020, greenestamps:
Answer by josgarithmetic(39630)   (Show Source): You can put this solution on YOUR website!
Needed to stay. It was good;
x=1

If find index went to 96, then x is 1.
Using formula, sum of arithmetic sequence,


Omitting steps, but leads to
If subtract x from both sides, then
.
.

NOT FINISHED...
Answer by ikleyn(52915)   (Show Source): You can put this solution on YOUR website!
.
1 + 4 + 7 + 10 + ... + x = 287
find x
~~~~~~~~~~~~~~~~~~~~~~~~


It is an anti-mathematical way to present a problem.

It makes me feel sick.

As well as any other person,  who has a  Math education and feeling of harmony . . .


Normal and standard  " human "  way  (= right way)  to formulate this problem mathematically is  THIS 

    The sum of the first n terms of an arithmetic progression
    with the first term 1 and the common difference 3 is 287.
    Find the number of terms "n" and the value of the n-th term.


How the problem is presented in your post is  .

It reminds me of how the witch doctors stir the potion in cups,
in  700  years old tales.



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answer: x = 40

Explanation

1, 4, 7, 10, ... is an arithmetic sequence
a1 = 1 is the first term
d = 3 is the common difference.

Sn = sum of the first n terms
Sn = 0.5n*(2*a1+d*(n-1))
287 = 0.5n*(2*1+3*(n-1))
1.5n^2-0.5n-287 = 0

Using the quadratic formula will yield solutions n = -13.667 (approximate) and n = 14.
We ignore the negative value of n.
n must be a positive whole number 1,2,3,etc
I'll let the student handle the scratch work for the quadratic formula.

We determined there are 14 terms being added in {1,4,7,10,...,x} to yield the sum 287.

Let's determine the 14th term.
an = a1 + d(n-1)
an = 1 + 3*(n-1)
an = 3n-2
a14 = 3*14-2
a14 = 40

Therefore, x = 40 and 1+4+7+10+...+40 = 287

Verification using WolframAlpha
Another way to confirm is to use a spreadsheet.

Answer by greenestamps(13216)   (Show Source): You can put this solution on YOUR website!


For an arithmetic sequence....

Sum = (number of terms)*(average of terms)

Number of terms: ((last term minus first term)/common difference)+1

=

Average of terms = average of first and last terms =

The sum is 287:











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A hint for doing that factoring.... We want to write



Since the linear coefficient is 3, a and b are "close together". So let them be "approximately equal". Then 1720 is "close to" a perfect square. The square root of 1720 is a bit more than 40; and since the product ab has final digit 0, one of a or b should be 40. Then a bit of arithmetic shows that a and b are in fact 43 and 40.

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Obviously ignore the negative solution....

ANSWER: x=40
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