Sum of infinite number of terms of GP is 20 and sum of their square is 100.

Question

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

Accepted Answer

Let $a=$ first team of $G.P.$ and $r=$ common ratio of $G.P.;$

Then $G.P.$ is $a,$ $ar,$ $a{r^2}$

Given $${S_\infty } = 20 \Rightarrow {a \over {1 - r}} = 20$$

$$ \Rightarrow a = 20\left( {1 - r} \right)....\left( i \right)$$

Also ${a^2} + {a^2}{r^2} + {a^2}{r^4} + ...$ to $$\infty  = 100$$

$$ \Rightarrow {{{a^2}} \over {1 - {r^2}}} = 100$$

$$ \Rightarrow {a^2} = 100\left( {1 - r} \right)\left( {1 + r} \right)....\left( {ii} \right)$$

From $(i),$ $${a^2} = 400{\left( {1 - r} \right)^2};$$

From $(ii),$ we get $$100\left( {1 - r} \right)\left( {1 + r} \right)$$

$$\,\,\,\,\,\,\,\,\,\, = 400{\left( {1 - r} \right)^2}$$

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

$$ \Rightarrow 5r = 3$$

$$ \Rightarrow r = 3/5.$$

Solutions · 1