Question:
The $4^{\text {th }}$ term of a G.P. is square of its second term, and the first term is $-3$. Determine its $7^{\text {th }}$ term.
Solution:
Let a be the first term and r be the common ratio of the G.P.
$\therefore a=-3$
It is known that, $a_{n}=a r^{n-1}$
$\therefore a_{4}=a r^{3}=(-3) r^{3}$
$a_{2}=a r^{1}=(-3) r$
According to the given condition,
$(-3) r^{3}=[(-3) r]^{2}$
$\Rightarrow-3 r^{3}=9 r^{2}$
$\Rightarrow r=-3$
$a_{7}=a r^{7-1}=a r^{6}=(-3)(-3)^{6}=-(3)^{7}=-2187$
Thus, the seventh term of the G.P. is $-2187$.
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