Question

If the $4$th and $8$th terms of a GP are $24$ and $384$ respectively, find the first term and common ratio.

Answer 1

We know that the formula for the $n$th term of an G.P is $T_n=a{r}^{n-1}$, where $a$ is the first term, $r$ is the common ratio.

It is given that the $4$th term is $T_4=24$, therefore,

$T_4=a{r}^{4-1}\Rightarrow 24=a{r}^3\Rightarrow a=\frac{24}{r^3}.....\left(1\right)$

It is also given that the $8$th term is $T_8=384$, therefore using equation 1,

$T_8=a{r}^{8-1}\Rightarrow 384=\frac{24}{r^3}\times r^7\Rightarrow \frac{384}{24}=r^4\Rightarrow r^4=16\Rightarrow r^4=2^4\Rightarrow r=2$

Now, substitute the value of $r$ in equation 1:

$a=\frac{24}{2^3}=\frac{24}{8}=3$

Hence, the first term is $3$ and the common ratio is $2$.