Question

If 3rd term of a G.P is 12 and 6th term is 96, then its 7th term will be

• A. $220$
• B. $150$
• C. $120$
• D. $192$

Answer 1

The correct option is D $192$

We know that, $n^{\text{th}}$ term of a G.P. is given by
$T_n=a\times r^{(n-1)},$
where $a=$ first terms, and $r=$ common ratio

Given, the third term is $12.$
$\Rightarrow T_3$ = 12
$\Rightarrow a{r}^{3-1}=12\Rightarrow a{r}^2=12\dots \left(i\right)$

and sixth term is $96$
$\Rightarrow T_6$ = 96
$\Rightarrow a{r}^{6-1}=96\Rightarrow a{r}^5=96\dots \left(ii\right)$

On dividing equation $\left(ii\right)$ by equation $\left(i\right)$, we get

$\Rightarrow \frac{a{r}^5}{a{r}^2}=\frac{96}{12}$
$\Rightarrow r^3=8$
$\Rightarrow r=\sqrt[3]{38}$
$\Rightarrow r=\sqrt[3]{32\times 2\times 2}$
$\Rightarrow r=2$

Putting $r=2$ in equation $\left(i\right)$, we get
$\Rightarrow a\times 2^2=12\Rightarrow 4a=12\Rightarrow a=\frac{12}{4}=3$

$\therefore \text{Seventh term}{T}_7=a{r}^{7-1}$
$=a{r}^6$
$=3\times 2^6$
$=3\times 64$
$=192$