Try this beautiful problem from the American Invitational Mathematics Examination, AIME 2012 based on Arithmetic Sequence.
The terms of an arithmetic sequence add to \(715\). The first term of the sequence is increased by \(1\), the second term is increased by \(3\), the third term is increased by \(5\), and in general, the \(k\)th term is increased by the \(k\)th odd positive integer. The terms of the new sequence add to \(836\). Find the sum of the first, last, and middle terms of the original sequence.
Series
Number Theory
Algebra
Answer: is 195.
AIME, 2012, Question 2.
Elementary Number Theory by David Burton .
After the adding of the odd numbers, the total of the sequence increases by \(836 - 715 = 121 = 11^2\).
Since the sum of the first \(n\) positive odd numbers is \(n^2\), there must be \(11\) terms in the sequence, so the mean of the sequence is \(\frac{715}{11} = 65\).
Since the first, last, and middle terms are centered around the mean, then \(65 \times 3 = 195\)
Hence option B correct.
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