First slide

Introduction to statistics

Question

The GM of the series $1, 2, 4, 8, 16, \dots, 2^{n}$ is

Easy

A

$2^{n + 1 / 2}$
B

$2^{n + 1}$
C

$2^{n / 2}$
D

$2^{n}$

Solution

Let G be the GM of $1, 2, 4, 8, 16, \dots, 2^{n}$ Then.
$G = {(1 \times 2 \times 4 \times 8 \times \dots \times 2^{n})}^{\frac{1}{n + 1}} = {(1 \times 2 \times 2^{2} \times 2^{3} \times \dots \times 2^{n})}^{\frac{1}{n + 1}}$
$\Rightarrow G = {(2^{1 + 2 + 3 + \dots +^{n}})}^{\frac{1}{n + 1}} = {(2^{\frac{n (n + 1)}{2}})}^{\frac{1}{n + 1}} = 2^{n / 2}$

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