Question

The $4^{th}$ term of a G.P. is square of its second term, and the first term is $-3$. Determine its $7^{th}$ term.

Answer 1

Let $a$ be the first term and $r$ be the common ratio of the given G.P.
$\therefore a=-3$
It is known that, $a^n={ar}^{n-1}$
$\therefore a_4=a{r}^3=(-3)r^3$ and $a_2=a{r}^1=(-3)r$
Now according to the given condition,
$(-3)r^3={\left[\right(-3\left)r\right]}^2\Rightarrow -3r^3=9r^2\Rightarrow r=-3\therefore a_7=a{r}^{7-1}=a{r}^6=(-3){(-3)}^6=-{\left(3\right)}^7=-2187$
Thus the seventh term of the G.P. is $-2187.$