Question

The sum of two products taken two at a ti ne of the numbers $1,2,2^2$, $2^3$,......$2^{n-2}$, $2^{n-1}$ is

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

Answer 1

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

The given series is $1,2,2^2,2^3,.....,2^{n-2},2^{n-1}$

Sum of product of $2$ taken at a time

$1\times 2+2\times 2^2+2^2\times 2^3+......+2^{n-2}\times 2^{n-1}=2+2^3+2^5+......2^{2n-3}$

Here $a=2r=2^2$

Sum$=a\left(\frac{r^n-1}{r-1}\right)=2\left[\frac{{\left(2^2\right)}^n-1}{2^2-1}\right]=\frac{2[2^n-1]}{4-1}$

$=\frac{2}{3}[2^{2n}-1]=\frac{1}{3}{2}^{2n}-\frac{1}{3}$