SOLUTION: Find the sum of the 1st 20 terms of the arithmetic sequence -2, -5, -8, -11

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Question 1196342: Find the sum of the 1st 20 terms of the arithmetic sequence
-2, -5, -8, -11
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the sum of the 1st 20 terms of the arithmetic sequence
-2, -5, -8, -11.
~~~~~~~~~~~~~~~~ 

First term is  a%5B1%5D = -2.

The common difference is  d = (-5) - (-2) = -5 + 2 = -3.


Use the formula for the sum of the first n term of an arithmetic progression

    S%5Bn%5D = %28%28a%5B1%5D+%2B+a%5Bn%5D%29%2F2%29%2An.


You need to know the 20th term  a%5B20%5D. It is (use the formula for the n-th term)

    a%5B20%5D = a%5B1%5D+%2B+%2820-1%29%2Ad = -2 + 19*(-3) = -59.


Therefore,  S%5B20%5D = %28%28-2%29+%2B+%28-59%29%29%2F2%29%2A20 = %28%28-61%29%2F2%29%2A20 = %28-61%29%2A10 = -610.


ANSWER.  The sum of the first 20 terms of this AP is  -610.

Solved.

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For introductory lessons on arithmetic progressions see
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Arithmetic progressions".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


To find the average of any set of numbers, you divide the sum of the numbers by the number of numbers in the set. So if you know the average of a set of numbers, you can find the sum by multiplying that average by the number of numbers in the set.

In this problem, we know there are 20 terms; to find the sum we need to find the average.

Finding the average of a set of numbers in arithmetic sequence is easy. Because of the equal spacing between terms, the average of all the terms is the average of the first and last terms.

We know the first term; and finding the last term is easy using the common difference.

The first term is -2; the common difference is -3; so the last (20th) term is the first term, plus the common difference 19 times: -2 plus 19(-3), or -59.

Then the average of all the terms is the average of the first and last terms: ((-2)+(-59))/2 = -61/2.

And finally the sum of all the terms is 20(-61/2) = -610.

ANSWER: -610