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 = ${\displaystyle \frac{a(r^3-1)}{r-1}}$

Now, sum of first six terms = S₆ = ${\displaystyle \frac{a(r^6-1)}{r-1}}$

It is given that

${\displaystyle \frac{\frac{a(r^3-1)}{r-1}}{\frac{a(r^6-1)}{r-1}}=\frac{125}{152}}$

${\displaystyle \Rightarrow \frac{r^3-1}{r^6-1}=\frac{125}{152}}$

${\displaystyle \Rightarrow \frac{r^3-1}{{\left(r^3\right)}^2-{\left(1\right)}^2}=\frac{125}{152}}$

${\displaystyle \Rightarrow \frac{r^3-1}{(r^3-1)(r^3+1)}=\frac{125}{152}}$

${\displaystyle \Rightarrow \frac{1}{r^3+1}=\frac{125}{152}}$

${\displaystyle \Rightarrow r^3+1=\frac{152}{125}}$

${\displaystyle \Rightarrow r^3=\frac{152}{125}-1}$

= ${\displaystyle \frac{152-125}{125}}$

= ${\displaystyle \frac{27}{125}}$

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

Hence, the common ratio is ${\displaystyle \frac{3}{5}}$.