# 1994 AHSME Problems/Problem 14

## Problem

Find the sum of the arithmetic series $$20+20\frac{1}{5}+20\frac{2}{5}+\cdots+40$$ $\textbf{(A)}\ 3000 \qquad\textbf{(B)}\ 3030 \qquad\textbf{(C)}\ 3150 \qquad\textbf{(D)}\ 4100 \qquad\textbf{(E)}\ 6000$

## Solution

### Brief Introduction

For those that do not know the formula, the sum of an arithmetic series with first term $a_1$, last term $a_n$ as $n$ terms, is $$S = \frac{n(a_1+a_n)}{2}.$$ We can prove this as follows:

Let $d$ be the common difference between terms of our series and let $n$ be the number of terms in our series. Let $a_1$ be the first term. Our series is $$a_1,~a_1+d,~a_1+2d,\dots,a_1+(n-1)d.$$ Note that we have $n-1$ in the last term because $a_1$ is a term. Let $S$ be our sum such that $$S=a_1+(a_1+d)+\dots+(a_1+(n-1)d).$$ We can rewrite our sums as $$S=(a_1+(n-1)d)+(a_1+(n-2)d)+\dots+(a_1+d)+a_1.$$ Adding these two sums together essentially creates $n$ pairs of $a_1+(a_1+(n-1)d)$ as shown below: $$2S=n(a_1+(a_1+(n-1)d))\implies S=\frac{n(a_1+a_n)}{2}$$ We use $a_n$ in place of $a_1+(n-1)d$ to represent the last term.

### Solving

Our first term is $20$ and our last term is $40$. To find the number of terms, $n$, we note that the common difference between each term is $\frac{1}{5}$. So we have $$20+\frac{1}{5}(n-1)=40\implies n-1=100\implies n=101.$$ Using our formula, our sum is $$101\left(\frac{20+40}{2}\right)=101\times 30=\boxed{\textbf{(B) }3030.}$$

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