Question

The $6$th term of the G.P is $32$ and its $8$th term is $128$, then the common ratio of the G.P is:

• A. $-1$
• B. $2$
• C. $4$
• D. $-4$

Answer 1

The correct option is B

$2$

Finding the common ratio:

Given that the $6$th term of the G.P is $32$ and its $8$th term is $128$

$a_6=32,a_8=128$.

We know that in G.P, the consecutive terms differ by a ratio of $r$.

Calculating $r$

$$\begin{gathered} a_6=a.r^{n-1} \\ 32=a.r^5\dots \dots \left(1\right) \\ {a}_8=a.r^7 \\ 128=a.r^7\dots \dots \left(2\right) \end{gathered}$$

Dividing equations $\left(2\right)\text{by}\left(1\right)$

$$\begin{gathered} \frac{a{r}^7}{a{r}^5}=\frac{128}{32} \\ \frac{r^7}{r^5}=4 \\ {r}^{(7-5)}=4 \\ {r}^2=4 \\ r=2 \end{gathered}$$

Hence, the common ratio is $2.$

Therefore, the correct answer is Option (B).