1968 AHSME Problems/Problem 17

Problem

Let $f(n)=\frac{x_1+x_2+\cdots +x_n}{n}$ , where $n$ is a positive integer. If $x_k=(-1)^k, k=1,2,\cdots ,n$ , the set of possible values of $f(n)$ is:

$\text{(A) } \{0\}\quad \text{(B) } \{\frac{1}{n}\}\quad \text{(C) } \{0,-\frac{1}{n}\}\quad \text{(D) } \{0,\frac{1}{n}\}\quad \text{(E) } \{1,\frac{1}{n}\}$

Solution

Because we start with $x_1=-1$ , and the terms $x_k$ alternate between $1$ and $-1$ , there is either one more $-1$ than the number of $1$ s (when $n$ is odd), or there are an equal number of $1$ s and $-1$ s (when $n$ is even). In the former case, $f(n)=\frac{-1}{n}$ , and, in the latter case, $f(n)=0$ . This is only consistent with answer $\fbox{C}$ .

1968 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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1968 AHSME Problems/Problem 17

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