Question

Find the arithmetic mean of $1,3,3^2,......,3^{n-1}$

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

Answer 1

The correct option is B $\frac{3^n-1}{2n}$

For finding the arithmetic mean of a given series, we take the sum of all the numbers in the series and divide that by the total number of terms.

Therefore arithmetic mean is given by $\frac{1+3+3^2+...+3^{n-1}}{n}$

Sum of n terms of a geometric progression with $a$ being the first term and $r$ as the common ratio is given by $\frac{a(r^n-1)}{r-1}$

Here, $a=1$ and $r=3$

$\therefore$ arithmetic mean $=\frac{1\times (3^n-1)}{(3-1)n}$

$\therefore$ arithmetic mean $=\frac{3^n-1}{2n}$