Question

If the sum of $n$ terms of a G.P. is $255$ and $n^{th}$ term is $128$ and common ratio is $2$, then first term will be

• A. $1$
• B. $3$
• C. $7$
• D. None of these

Answer 1

The correct option is A

$1$

Explanation for the correct option.

The $n^{th}$ term of G.P. $\mathbf{=}{\mathbf{ar}}^{\mathbf{n}\mathbf{-}\mathbf{1}}$

Here, $a=\mathrm{first}\mathrm{term}\mathrm{and}r=2$

$$\begin{gathered} \Rightarrow 128=a\left(2^{n-1}\right) \\ \Rightarrow 128=a\frac{2^n}{2} \\ \Rightarrow 256=a\left(2^n\right)................\left(1\right) \end{gathered}$$

Sum of $n$ terms of G.P. $\mathbf{=}{\mathit{S}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{a}\left(r^n-1\right)}{\mathbf{r}\mathbf{-}\mathbf{1}}$

$$\begin{gathered} \Rightarrow 255=\frac{a\left(2^n-1\right)}{2-1} \\ \Rightarrow 255=a\left(2^n\right)-a \\ \Rightarrow 255=256-a\text{from}\left(1\right) \\ \Rightarrow a=1 \end{gathered}$$

Hence, option A is correct.