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  • Need clarity, kindly explain! Sum of infinite number of terms in GP is 20 and sum of their square is 100. The common ratio of GP is

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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