Difference between revisions of "1994 AHSME Problems/Problem 14"

Revision as of 17:35, 9 January 2021

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 $n=\frac{a_1+a_n}{2}.$ 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 $S=a_1+(a_1+d)+\dots+(a_1+(n-d)d).$ 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: $2S=n(a_1+(a_1+(n-1)d))\implies S=n\left(\frac{a_1+a_n}{2}\right).$ 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 $101\left(\frac{20+40}{2}\right)=101\times 30=\boxed{\textbf{(B) }3030.}$

--Solution by TheMaskedMagician

1994 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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 --Solution by [http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=200685 TheMaskedMagician]
+==See Also==
+{{AHSME box|year=1994|num-b=13|num-a=15}}
+{{MAA Notice}}

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