Question

The first term of a G.P whose second term is$2$ and sum to infinity is$8$, will be

• A. $6$
• B. $3$
• C. $4$
• D. $1$

Answer 1

The correct option is C

$4$

The explanation for the correct answer.

Solve for the first term of the G.P.

The second term of G.P. is$2$

$ar=2,r=\frac{2}{a}$

Sum of infinity $\mathbf{=}\frac{\mathbf{a}}{\mathbf{1}\mathbf{-}\mathbf{r}}$

$a=$the first term

$r=$the common ratio

$$\begin{gathered} \therefore \frac{a}{1-\left({\displaystyle \frac{2}{a}}\right)}=8 \\ \Rightarrow 8\left(1-\frac{2}{a}\right)=a \\ \Rightarrow 8a-16=a^2 \\ \Rightarrow a^2-8a+16=0 \\ \Rightarrow {(a-4)}^2=0\Rightarrow a=4 \end{gathered}$$

So, if the second term is $2$ and the sum to infinity is $8$ then the first term of a G.P. will be $4$

Hence, option (C) is the correct answer.