Question

Find $8th$ and $12$th terms of the G.P. $81,-27,9,...........$

Answer 1

It is given that the first term of G.P is $a_1=81$ and the second term is $a_2=-27$, therefore,

$r=\frac{a_2}{a_1}=\frac{-27}{81}=-\frac{1}{3}$

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 first term is $a=81$ and the common ratio is $r=-\frac{1}{3}$, thus,

$T_8=81\times {(-\frac{1}{3})}^{8-1}=81\times {(-\frac{1}{3})}^7=81\times (-\frac{1}{3^7})=-81\times \frac{1}{2187}=-\frac{1}{27}$

$T_{12}=81\times {(-\frac{1}{3})}^{12-1}=81\times {(-\frac{1}{3})}^{11}=81\times (-\frac{1}{3^{11}})=-81\times \frac{1}{177147}=-\frac{1}{2187}$

Hence, $T_8=-\frac{1}{27}$ and $T_{12}=-\frac{1}{2187}$.