SOLUTION: the sum of the first 20 terms of an AP series is identical to the sum of the first 22 terms. If the common difference is -2 find the first term?

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Question 1128870: the sum of the first 20 terms of an AP series is identical to the sum of the first 22 terms. If the common difference is -2 find the first term?
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
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The condition immediately implies that the sum 

    a%5B21%5D + a%5B22%5D = 0.


Thus  a%5B22%5D = -a%5B21%5D.


From the other side,  a%5B22%5D = a%5B21%5D+%2B+d = -a%5B22%5D+%2B+%28-2%29.


It implies  2%2Aa%5B22%5D = -2;  hence,  a%5B22%5D = -1.



From the other side,  a%5B22%5D = a%5B1%5D+%2B+21%2Ad = a%5B1%5D+%2B+21%2A%28-2%29 = a%5B1%5D+-+42.


Hence,  a%5B1%5D = a%5B22%5D+%2B+42 = -1 + 42 = 41.


Answer.  a%5B1%5D = 41.

Solved.

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    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
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