1984 AIME Problems/Problem 1
Problem
Find the value of if
,
,
is an arithmetic progression with common difference 1, and
.
Solution
One approach to this problem is to apply the formula for the sum of an arithmetic series in order to find the value of , then use that to calculate
and sum another arithmetic series to get our answer.
A somewhat quicker method is to do the following: for each , we have
. We can substitute this into our given equation to get
. The left-hand side of this equation is simply
, so our desired value is
.