SOLUTION: What is the first of 100 consecutive odd integers whose sum is 100^100?

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Question 1191256: What is the first of 100 consecutive odd integers whose sum is 100^100?
Answer by ikleyn(52803) About Me  (Show Source):
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What is the first of 100 consecutive odd integers whose sum is 100^100?
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Let the first term be  a%5B1%5D.


Then the 100-th term is  a%5B1%5D%2B2%2A99.


The average of the first and the 100-th term is  %28a%5B1%5D%2B%28a%5B1%5D%2B2%2A99%29%29%2F2 = a%5B1%5D%2B99.


The sum of this sequence must be  100%5E100,  which gives this equation


    %28a%5B1%5D%2B99%29%2A100 = 100%5E100.


It implies  a%5B1%5D = 100%5E99+-+99.    ANSWER

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".


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

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