If the sum of an infinite G.P. and the sum of | vedclass

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Question

(b) Let the first series be $a + ar + a{r^2} + .........$

then the second series is ${a^2} + {a^2}{r^2} + {a^2}{r^4} + ..........$

their sums ar

Vedclass · Accepted Answer

$\frac{1}{2}$

Vedclass · Answer

(b) Let the first series be $a + ar + a{r^2} + .........$

then the second series is ${a^2} + {a^2}{r^2} + {a^2}{r^4} + ..........$

their sums are given as $3$. So, we have

$\frac{a}{{1 - r}} = 3$ or $a = 3(1 - r)$

and$\frac{a^2}{{1 - r^2}} = 3$ or ${a^2} = 3(1 - {r^2})$

Eliminating $a,\;{\left\{ {3\,(1 - r)} \right\}^2} = 3\,(1 - {r^2})$

$ \Rightarrow $$3\,(1 - r) = (1 + r)$,

$ \Rightarrow $$4r = 2$ or $r = \frac{1}{2}$.

If the sum of an infinite $G.P.$ and the sum of square of its terms is $3$, then the common ratio of the first series is

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