Question

The geometric mean of $1,2,2^2,.....,2^n$ is

• A. $2^{\frac{n}{2}}$
• B. $n^{\frac{n+1}{2}}$
• C. $2^{\frac{n(n+1)}{2}}$
• D. $2^{\frac{n-1}{2}}$

Answer 1

The correct option is A

$2^{\frac{n}{2}}$

Explanation for the correct option:

Compute the geometric mean of the given geometric progression.

A geometric progression $1,2,2^2,.....,2^n$ is given.

It is clear that the given geometric progression has $n+1$ terms.

So, the Geometric mean $G.M$ is as follows:

$$\begin{gathered} G.M={\left(1·2·2^2·....·2^n\right)}^{\frac{1}{n+1}} \\ \Rightarrow G.M={\left(2^{1+2+3+...+n}\right)}^{\frac{1}{n+1}} \\ \Rightarrow G.M={\left(2^{\frac{n(n+1)}{2}}\right)}^{\frac{1}{n+1}} \\ \Rightarrow G.M=2^{\frac{n}{2}} \end{gathered}$$ [$\because$Since the sum of $n$ natural numbers is $\frac{n(n+1)}{2}$]

Therefore, the geometric mean of the given geometric progression is $2^{\frac{n}{2}}$.

Hence, option $A$ is the correct option.