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 $\fbox{C}$.

See also

1968 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
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 31 32 33 34 35
All AHSME Problems and Solutions

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