1994 AHSME Problems/Problem 14
Problem
Find the sum of the arithmetic series
Solution
Brief Introduction
For those that do not know the formula, the sum of an arithmetic series with first term , last term
as
terms, is
We can prove this as follows:
Let be the common difference between terms of our series and let
be the number of terms in our series. Let
be the first term. Our series is
Note that we have
in the last term because
is a term. Let
be our sum such that
We can rewrite our sums as
Adding these two sums together essentially creates
pairs of
as shown below:
We use
in place of
to represent the last term.
Solving
Our first term is and our last term is
. To find the number of terms,
, we note that the common difference between each term is
. So we have
Using our formula, our sum is
--Solution by TheMaskedMagician