Problem:
he terms of an arithmetic sequence add to . The first term of the sequence is increased by , the second term is increased by , the third term is increased by , and in general, the th term is increased by the th odd positive integer. The terms of the new sequence add to . Find the sum of the first, last, and middle terms of the original sequence.
Solution:
If the original sequence has terms, then the sum of the added values is , and . It follows that is times the average value of the terms of the original sequence. This average, which is , is the value of the middle term. The first and last terms of the sequence must add to twice , so the sum of the first, last, and middle terms is .
The problems on this page are the property of the MAA's American Mathematics Competitions