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 $5$ and determine $T_1,T_3,T_5$ of $G.P.$.

Answer 1

$a=5,a{r}^2=a+3d,a{r}^4=a+15d$
$\therefore 5r^2=5+3d,5r^4=5+15d$
or $r^4=1+3d$ Multiply by $25$
$25r^4=25+75d$
or ${(5+3d)}^2=25+75d$
or $25+30d+9d^2=25+75d$
$\therefore 9d^2-45d=0$ $\therefore d=5,0$.
$\therefore T_4=a+3d=5+15=20=a{r}^2:5r^2$
$\therefore r=2,-2$
$\therefore 5,20,80$ are the $1st$, $3rd$ and $5th$ terms of $G.P$. And $T_4$ of $A.P.=20$