jmc

Question

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 kth term is increased by the kth 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.

Accepted Answer

The sum of all the increases is given by 1 + 3 + 5 + + (2k-1) = k^2.Thus 715 + k^2 = 836, or k^2 = 121, so k = 11. Then the middle term of the sequence must be 71511 = 65. Since the original sequence is arithmetic, the sum of the first, last, and middle term is simply 3 65 = 195.