Question

The sum.of infinite geometric series is 20 .when all the sum in the series are squared,the sum of resulting series is 80

Answer 1

Its pretty simple.
Please see image

Answer 2

Answer: r = 2/3

The sum of infinite series in G.P  

$s1\quad =\quad a+ar+ar\times 2+ar\times 3--------= \frac { a }{ (1-r) }$

$\frac { a }{ (1-r) } \quad =\quad 20- -- - - -- - - - - - (1)$

$s^2\quad =\quad { a }^{ 2 }+{ a }^{ 2 }{ r }^{ 2 }+{ a }^{ 2 }{ r }^{ 4 }+{ a }^{ 2 }{ r }^{ 8 }--------= \frac { a }{ (1-r) }$

$80\quad =\quad { a }^{ 2 }(1+{ r }^{ 2 }+{ r }^{ 4 }+-------)$

$80\quad =\quad { a }^{ 2 }(\frac { 1 }{ (1-{ r }^{ 2 }) } )- - - - - - - - (2)$

Dividing equation square of (1)/(2) , we get  

$\frac { 400 }{ 80 } \quad =\quad \frac { 1+r }{ 1-r }$

$5\quad =\quad \frac { 1+r }{ 1-r }$

1+r = 5 – 5r  

6r = 4  

$r\quad =\quad \frac { 2 }{ 3 }$