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  4. An infinite G.P. has first term as a and sum as 5 , then

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Q. An infinite G.P. has first term as $a$ and sum as $5$ , then

Sequences and Series

Solution:

$S_{\infty}=\frac{a}{1-r}=5($ given $)$
$\Rightarrow r=\frac{5-a}{5}$
But $0<|r|<1$
$\Rightarrow 0<\left|\frac{5-a}{5}\right|<1$
$\Rightarrow -1<\frac{5-a}{5}< 1$ and $a \neq 5$
$\Rightarrow -5 < 5-a < 5$ and $a \neq 5$
$\Rightarrow -10 < -a < 0$ and $a \neq 5$
$\Rightarrow 10 > a > 0$ and $a \neq 5$
$\Rightarrow 0< a< 10$ and $a \neq 5$