Let \beginaligned S_n&=1-2+3-4+\cdots\ &\quad+(-1)^n-1n. \endaligned Here n=1, 2, …. Then S_17+S_33+S_50 equals:

1968 AMC 12 Problem 27 — Solution

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27.

Let

$\begin{aligned} S_n&=1-2+3-4+\cdots\\ &\quad+(-1)^{n-1}n. \end{aligned}$

Here $n=1,$ $2,$ $\ldots.$ Then $S_{17}+S_{33}+S_{50}$ equals:

$0$

$1$

$2$

$-1$

$-2$

Pair terms as $(1-2)+(3-4)+\cdots$

For even $n,$ $S_n=-\frac{n}{2};$ for odd $n,$ $S_n=\frac{n+1}{2}$

Pairing gives $S_n=-\frac{n}{2}$ for even $n$ and $S_n=\frac{n+1}{2}$ for odd $n.$ Hence $S_{17}=9,$ $S_{33}=17,$ $S_{50}=-25,$ and their sum is $1.$

Therefore, the correct answer is B.