Question

If the ${10}^{th}$ term of a G.P. is $9$ and $4^{th}$ term is $4$, then its $7^{th}$ term is

• A. $6$
• B. $14$
• C. $\frac{27}{14}$
• D. $\frac{56}{15}$

Answer 1

Answer: (A)

$6$

The terms of a $G.P$ are form $a,ar,a{r}^2,a{r}^3,.....$

The ${10}^{th}$ term of a $G.P$ is $9$

$\Rightarrow$ $a{r}^9=9$

$\Rightarrow$ $r^9=\frac{9}{a}$ ----- ( 1 )

The $4^{th}$ term of a $G.P$ is $4$

$\Rightarrow$ $a{r}^3=4$ ---- ( 2 )

Cubing both sides,

$\Rightarrow$ $a^3{r}^9=64$

$\Rightarrow$ $a^3\times \frac{9}{a}=64$ [ From ( 1 ) ]

$\Rightarrow$ $9a^2=64$

$\Rightarrow$ $a^2=\frac{64}{9}$

$\Rightarrow$ $a=\frac{8}{3}$

Substituting value of $a$ in ( 2 ),

$\Rightarrow$ $\frac{8}{3}\times r^3=4$

$\Rightarrow$ $r^3=\frac{12}{8}$

$\Rightarrow$ $7^{th}$ term $=a{r}^6$

$=\frac{8}{3}\times (r^3{)}^2$

$=\frac{8}{3}\times {\left(\frac{12}{8}\right)}^2$

$=\frac{8}{3}\times \frac{144}{64}$

$=6$