Question

The $4^{\mathrm{th}}$ term of a GP is $16$and $7^{\mathrm{th}}$ term is $128$ .Find the first term and common ratio of the series.

Answer 1

Step 1: Find the common ratio of a GP

Let the first term be $a$ and the common ratio be $r$ of a GP

Here it is given that $4^{\mathrm{th}}$ term $16$and $7^{\mathrm{th}}$ term is $128$

$$\begin{gathered} a_4=a{r}^3=16....\left(i\right) \\ {a}_7=a{r}^6=128....\left(ii\right) \end{gathered}$$

$\left(\mathrm{ii}\right)÷\left(\mathrm{i}\right)$

$\frac{a{r}^6}{a{r}^3}=\frac{128}{16}$

$\Rightarrow r^3=8$

$\Rightarrow r=2$

Step 2. Find the first term of a GP:

Now,

$\begin{array}{rcl}& & \begin{array}{cc}\Rightarrow & a{r}^3=16\\ \Rightarrow & a\times 2^3=16\\ \Rightarrow & a=\frac{16}{8}\\ \therefore & a=2\end{array}\end{array}\begin{array}{}\end{array}$

Therefore, the first term is $2$and the common ratio is $2$.