Write the sum to n terms of a series whose r^th term is r + 2^r.

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Solution

Series whose rth term is r + 2^r:

$(1 + 2^{1}) + (2 + 2^{2}) + (3 + 2^{3}) + (4 + 2^{4}) + . . . + (n + 2^{n})$

Thus, we have:

$S_{n} = (1 + 2^{1}) + (2 + 2^{2}) + (3 + 2^{3}) + (4 + 2^{4}) + . . . + (n + 2^{n}) = (1 + 2 + 3 + 4 + . . . + n) + (2 + 2^{2} + 2^{3} + 2^{4} + . . . + 2^{n}) = \frac{n (n + 1)}{2} + 2 (\frac{2^{n} - 1}{2 - 1}) = \frac{n (n + 1)}{2} + 2^{n + 1} - 2$

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