Question

The geometric mean of the series $1,2,4,8,\cdots 2^n$is

• A. $2^{n/2}$
• B. $2^{n/4}$
• C. $2^{2n}$
• D. none of these

Answer 1

The correct option is A $2^{n/2}$

Given series is $1,2,4,8,\cdots 2^n$
Geometric mean of $a_1,a_2,a_3...a_n$ is ${(a_1{a}_2{a}_3,...a_n)}^{\frac{1}{n}}$
$\therefore$ GM of given series is ${\left({\mathrm{1.2.4...2}}^n\right)}^{\frac{1}{n+1}}$
$\to [2^{1+2+...+n}{]}^{\frac{1}{n+1}}$
$\to [2^{\frac{n(n+1)}{2}}{]}^{\frac{1}{n+1}}$
Hence, Geometric mean of the series is $2^{n/2}$