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Sum of infinite number of terms in GP is 20 and sum of their square is 100. The common ratio of GP is

Option 1)
$5$

Option 2)
$3/5$

Option 3)
$8/5$

Option 4)
$1/5$

Answers (1)

As we learnt in

Sum of infinite terms of a GP -

$a+ar+ar^{2}+- - - - -= \frac{a}{1-r}\\here \left | r \right |<1$

- wherein

$a\rightarrow$ first term

$r\rightarrow$ common ratio

Let the GP be

$a_{1}ar_{1}ar^{2}\cdot \cdot \infty$

$S_{\infty }=\frac{a}{1-r}=20$

$\Rightarrow \left ( \frac{a}{1-r} \right )^{2}= 400$ (i)

Similarly

$S_{\infty }= \frac{a^{2}}{1-r^{2}} = 100$ (ii)

$\frac{eg \left ( i\right )}{eg\left ( ii \right )}= \frac{a^{2}}{\left ( 1-r \right )^{2}}\times \frac{1-r^{2}}{a^{2}}= \frac{400}{100}= 4$

$= \frac{\left ( 1-r \right )\left ( 1+r \right )}{\left ( 1-r \right )\left ( 1-r \right )}= 4$

$\Rightarrow 1+r= 4-4r$

$5r=3$

$r= \frac{3}{5}$

Option 1)

$5$

This option is incorrect.

Option 2)

$3/5$

This option is correct.

Option 3)

$8/5$

This option is incorrect.

Option 4)

$1/5$

This option is incorrect.

Posted by

Vakul

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