Question

The G.M of the numbers $3,3^2,3^3,.....3^n$ is

• A. $3^{\frac{2}{n}}$
• B. $3^{\frac{n+1}{2}}$
• C. $3^{\frac{n}{2}}$
• D. $3^{\frac{n-1}{2}}$

Answer 1

The correct option is B

$3^{\frac{n+1}{2}}$

Explanation for the correct option:

Compute the geometric mean of the given geometric progression.

A geometric progression $3,3^2,3^3,.....3^n$ is given.

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

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

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

Therefore, the geometric mean of the given geometric progression is $3^{\frac{n+1}{2}}$.

Hence, option $B$ is the correct option.