The arithmetic mean of the numbers 1,3,32,…,3n−1, is

The arithmetic mean of the numbers $1, 3, 3^{2}, \dots, 3^{n - 1},$ is

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

Solution:

Let M be the required mean. Then,

$M = \frac{1 + 3 + 3^{2} + \dots + 3^{n - 1}}{n}$

$\Rightarrow M = \frac{1 \times (\frac{3^{n} - 1}{3 - 1})}{n} = \frac{3^{n} - 1}{2 n}$

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