Question

In a G.P., the ratio between the sum of first three terms and that of the first six terms is 125 : 152. Find its common ratio.

Answer 1

Let a be the first term and r be the common ratio of given G.P.

Now, sum of first three terms = S₃ = `(a(r^3 - 1))/(r - 1)`

Now, sum of first six terms = S₆ = `(a(r^6 - 1))/(r - 1)`

It is given that

`((a(r^3 - 1))/(r - 1))/((a(r^6 - 1))/(r - 1)) = 125/152`

`=> (r^3 - 1)/(r^6 - 1) = 125/152`

`=> (r^3 - 1)/((r^3)^2 - (1)^2) = 125/152`

`=> (r^3 - 1)/((r^3 - 1)(r^3 + 1)) = 125/152`

`=> 1/(r^3 + 1) = 125/152`

`=> r^3 + 1 = 152/125`

`=> r^3 = 152/125 - 1`

= `(152 - 125)/125`

= `27/125`

`=> r = 3/5`

Hence, the common ratio is `3/5`.