Question

Find the sum of n terms of a series whose mth term is $$2^m + 2m$$.

[Hint: $$S_n = \left[(2^1 + 2 \times 1) + (2^2 + 2 \times 2) + (2^3 + 2 \times 3) +\ .......\right]$$

$$= \left[(2^1 + 2^2 + 2^3 +\ ........ + 2^ \ .........\ +\ 2 \times 1 + 2 \times 2 + 2 \times 3 +\ ....)\right]$$

$$= 2^1 + 2^2 + 2^3 +\ ........ + 2(1 + 2 + 3 +\ ......)$$]

Answer 1

Using the hint:

$$S_n = (2^1 + 2^2 + 2^3 + \ldots + 2^n) + 2(1 + 2 + 3 + \ldots + n)$$

The first part is a G.P. with $$a = 2$$ and $$r = 2$$, so

$$2^1 + 2^2 + \ldots + 2^n = \frac{2(2^n - 1)}{(2 - 1)} = 2^{n+1} - 2$$

The second part is $$2 \times \dfrac{n(n + 1)}{2} = n(n + 1)$$.

$$S_n = (2^{n + 1} - 2) + n(n + 1)$$

Hence, $$S_n = 2^{n + 1} + n(n + 1) - 2 = 2^{n + 1} + n^2 + n - 2$$.