### The arithmetic mean of the series 1, 2, 4, 8, 16, . . . . . . , $2^{n}$ is.

A. $\Large\frac{2^{n}-1}{n+1}$ B. $\Large\frac{2^{n}+1}{n}$ C. $\Large\frac{2^{n}-1}{n}$ D. $\Large\frac{2^{n+1}-1}{n+1}$ Answer: Option D
The given series is a G.P. with first term, a = 1 and common ratio, r = 2, It has (n + 1) terms. ∴ Sum of the terms of the series $\begin{array}{l}=\Large\frac{2^{n+1}-1}{2-1}\\ =2^{n+1}-1\\ \text{ Arithmetic mean }\\ =\Large\frac{2^{n+1}-1}{n+1}\end{array}$