Infinite Sequence

Question

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837

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+148

The sum of a certain infinite geometric series is 2. The sum of the squares of all the terms is 3. Find the common ratio.

noobieatmath Jan 31, 2019

edited by noobieatmath Jan 31, 2019

Rom · Answer 1 · 2019-02-01T09:28:21

$\sum \limits_{k=0}^\infty~a r^k = a \dfrac{1}{1-r}=2\\ a = 2(1-r)\\ \sum \limits_{k=0}^\infty~a^2 r^{2k} = \dfrac{a^2}{1-r^2}=3\\ a^2 = 3(1-r^2)$

$4(1-r)^2 = 3(1-r^2)\\ 7 r^2-8 r+1\\ \left(r-\dfrac 1 7\right)(r-1)\\ r = \dfrac 1 7 \text{ (1 is an extraneous solution)}$

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