Question

Find the sum to n terms of
$3.2+{5.2}^2+{7.2}^3+...$

• A. $2^{n+2}-2^{n+1}-2$
• B. $n{.2}^{n+2}-2^{n+1}+2$
• C. $n.2^{n+2}+2^{n-2}$
• D. None of these

Answer 1

The correct option is B $n{.2}^{n+2}-2^{n+1}+2$

Best way to solve this type of questions is to check options by putting n = 1, 2...
For n = 1, Sum = 6.
For n = 2, Sum = 6 + 20 = 26.
Putting values of n in option (b)
When n = 1, ${1.2}^{1+2}-2^{1+1}+2=6$
When n = 2, ${2.2}^{2+2}-2^{2+1}+2=26$
Hence, option (b) is correct answer.

Alternatively:
Let $S_n=3.2+{5.2}^2+{7.2}^3+...+(2n+1).2^n\therefore 2S_n={3.2}^2+{5.2}^3+...+(2n-1){.2}^n+(2n+1){.2}^{n+1}$
subtracting, we get
$-S_n=3.2+{2.2}^2+{2.2}^3+...+{2.2}^n-(2n+1){.2}^{n+1}=3.2+2(2^2+2^3+...+2^n)-(2n+1){.2}^{n+1}=etc$
simplifying it, we get option (b) as the answer.