Question

In a G.P. the first, third and fifth terms may be considered as the first,fourth and sixteenth terms of an A.P. Determine the fourth term of the A.P., knowing that its first term is 5 ?

Answer 1

According to the given information, $a=5,a{r}^2=5+3d,a{r}^4=5+15d$
$a$ is the first term, $r$ is the common ratio, $d$ is the common difference.
Thus, $r^2=1+\frac{3d}{5},r^4=1+3d$
So, $1+3d=1+\frac{6d}{5}+\frac{9d^2}{25}$
So,$\frac{9d^2}{25}=\frac{9d}{5}$
$\Rightarrow d=5$ and therefore, the fourth term is $5+3d=20$