Question

Fourth and seventh terms of a G.P. are ${\displaystyle \frac{1}{18}}$ and ${\displaystyle -\frac{1}{486}}$ respectively. Find the G.P.

Answer 1

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

4^th term =${\displaystyle \frac{1}{18}\Rightarrow a{r}^3=\frac{1}{18}}$

7^th term =${\displaystyle \frac{1}{486}\Rightarrow a{r}^6=\frac{1}{486}}$

Now, ${\displaystyle \frac{a{r}^6}{a{r}^3}=\frac{\frac{-1}{486}}{\frac{1}{18}}}$

${\displaystyle \Rightarrow r^3=-\frac{1}{27}}$

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

${\displaystyle a{r}^3=\frac{1}{18}}$

${\displaystyle \Rightarrow a\times {(-\frac{1}{3})}^3=\frac{1}{18}}$

${\displaystyle \Rightarrow a=-\frac{27}{18}=-\frac{3}{2}}$

∴ G.P. = a, ar, ar², ar³, .......

= ${\displaystyle -\frac{3}{2},-\frac{3}{2}\times (-\frac{1}{3}),-\frac{3}{2}\times {(-\frac{1}{3})}^2,\frac{1}{18},.......}$

= ${\displaystyle -\frac{3}{2},\frac{1}{2},-\frac{1}{6},\frac{1}{18},.......}$