Question

If mean of $n$ items is $\overline{x}$ ,if each item is successively increased by $3,3^2,3^3...{.3}^n,$ then new means equals

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

Answer 1

The correct option is B $\overline{x}+3\frac{(3^n-1)}{2n}$

Let n items be denoted by $x_1,x_2,x_3,...x_n,$
$\therefore$ new items are $x_1+3,x_2+3^2,x_3+3^3,...x_n+3^n$
$\therefore$ new mean $=\frac{(x_1+3)+(x_2+3^2)+(x_3+3^3+....+(x_n+3^n))}{n}$
$=\frac{(x_1+x_2...+x_n)}{n}+\frac{3^1+3^2+....+3^n}{n}$ $=\overline{x}+\frac{3(3^n-1)}{2n}$
using $S_n$ of G.P., as
$S_n$ $=\alpha \left(\frac{r^n-1}{r-1}\right),(r>1)$