Question

$2+3.2+{4.2}^2+\dots +n{.2}^{n-2}$ terms (for $n\ge 2$) $=$

• A. $n{.2}^{n+1}$
• B. $(n+1)2^n$
• C. $(n-1)2^{n-1}$
• D. $n{.2}^n$

Answer 1

The correct option is C $(n-1)2^{n-1}$

Let $S=2+3.2+{4.2}^2+{5.2}^3+{6.2}^4+.........n{2}^{n-2}$ ...(i)

Multiplying both sides by 2, we get

$2S=2.2+{3.2}^2+{4.2}^3+{5.2}^4+{6.2}^5+.........+(n-1)2^{n-2}+n{2}^{n-1}$ ...(ii)
Subtracting (i) from (ii), we get

$-S=2+2+2^2+2^3+.........+2^{n-2}-n{2}^{n-1}$

$-S=2+2\left(\frac{2^{n-2}-1}{2-1}\right)-n{2}^{n-1}$

$S=(n-1)2^{n-1}$
Hence, option 'C' is correct.