AHSME 1959 P49

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Problem Statement

For the infinite series $1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$ let $S$ be the (limiting) sum. Then $S$ equals:

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Solution

$$S = 1 - \dfrac{1}{2} - \dfrac{1}{4} + \dfrac{1}{8} - \dfrac{1}{16} - \dfrac{1}{32} +... $$ $$S = (1 + \dfrac{1}{8} + \dfrac{1}{64} + \dots) - (\dfrac{1}{2} + \dfrac{1}{16} + \dfrac{1}{64} + \dots) - (\dfrac{1}{4} + \dfrac{1}{32} + \dfrac{1}{256} + \dots)$$ $$S = \dfrac{1}{1 - \dfrac{1}{8}} - \dfrac{\dfrac{1}{2}}{1 - \dfrac{1}{8}} - \dfrac{\dfrac{1}{4}}{1 - \dfrac{1}{8}}$$ $$S = \dfrac{8}{7} - \dfrac{4}{7} - \dfrac{2}{7}$$ $$S = \dfrac{2}{7}$$