Given-that-the-series-x-3-y-3-2-x-3-3-y-3-4-x-3-n-1-y-3-n-8-x-y-R-Find-the-value-of-3x-y-

Question Number 143596 by ZiYangLee last updated on 16/Jun/21

Given that the series

\frac{x}{3^{}} + \frac{y}{3^{2}} + \frac{x}{3^{3}} + \frac{y}{3^{4}} + \dots + \frac{x}{3^{n - 1}} + \frac{y}{3^{n}} = 8, x, y \in R .

Find the value of 3 x + y .

Answered by bobhans last updated on 16/Jun/21

S_{1} = x (\frac{1}{3} + \frac{1}{3^{3}} + \frac{1}{3^{5}} + \dots) = x (\frac{1 / 3}{8 / 9}) = \frac{3 x}{8}

S_{2} = y (\frac{1}{3^{2}} + \frac{1}{3^{4}} + \frac{1}{3^{6}} + \dots) = y (\frac{1 / 9}{8 / 9}) = \frac{y}{8}

\Rightarrow \frac{3 x + y}{8} = 8 \Rightarrow 3 x + y = 64

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