Question

If the product of the first four consecutive terms of a G.P is $256$ and if the common ratio is $4$ and the first term is positive, then its $3^{rd}$ term is

• A. $8$
• B. $\frac{1}{16}$
• C. $\frac{1}{32}$
• D. $16$

Answer 1

The correct option is A $8$

Let the first four terms terms of the geometric progression be $a,ar,a{r}^2,a{r}^3$

It is given that the product of the first four consecutive terms is $256$, therefore,

$a\times ar\times a{r}^2\times a{r}^3=256\Rightarrow a^4{r}^6=256\Rightarrow a^4=\frac{256}{r^6}\Rightarrow a^4=\frac{256}{4^6}(Givenr=4)$

$\Rightarrow a^4=\frac{2^8}{2^{12}}\Rightarrow a^4=\frac{1}{2^4}\Rightarrow a^4={\left(\frac{1}{2}\right)}^4\Rightarrow a=\frac{1}{2}$

We know that the general term of an geometric progression with first term $a$ and common ratio $r$ is $T_n=a{r}^{n-1}$

To find the $3$th term of the G.P, substitute $n=3$, $a=\frac{1}{2}$ and $r=4$ in $T_n=a{r}^{n-1}$ as follows:

$T_3=a{r}^2=\frac{1}{2}\times (4{)}^{3-1}=\frac{1}{2}\times (4{)}^2=\frac{1}{2}\times 16=8$

Hence, the $3$rd term is $8$.