Question

The sum of the products taken two at a time of the numbers $1,2,2^2,\dots .2^{n-2},2^{n-1}$ is

• A. $\left(\frac{1}{3}\right)2^{2n}–2^{2n}-2^n+\left(\frac{2}{3}\right)$
• B. $\frac{1}{3}{2}^{2n}+2^{2n}-\frac{2}{3}$
• C. $–\left(\frac{1}{3}\right)2^{2n}+2^n-\frac{2}{3}$
• D. None of these

Answer 1

The correct option is D

None of these

Find the sum of the given series

$1,2,2^2,\dots .2^{n-2},2^{n-1}$

The sum of the products taken two at a time $=1\times 2+2\times 2^2+2^2\times 2^3+\dots .+2^{n-2}\times 2^{n-1}$

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

This is a G.P.with,

$a=2,r=2^2$

The sum of the GP series is

$$\begin{gathered} =\frac{a(r^n-1)}{r-1} \\ =\frac{2(2^{2n}-1)}{2^2-1} \\ =\frac{2}{3}(2^{2n}-1) \end{gathered}$$

Hence, the correct answer is $\mathrm{Option}\left(\mathrm{D}\right)$.