Question

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

 

Answer 1

Series whose rth term is r + 2^r: 

\[\left( 1 + 2^1 \right) + \left( 2 + 2^2 \right) + \left( 3 + 2^3 \right) + \left( 4 + 2^4 \right) + . . . + \left( n + 2^n \right)\]

Thus, we have: 

\[S_n = \left( 1 + 2^1 \right) + \left( 2 + 2^2 \right) + \left( 3 + 2^3 \right) + \left( 4 + 2^4 \right) + . . . + \left( n + 2^n \right)\]
\[ = \left( 1 + 2 + 3 + 4 + . . . + n \right) + \left( 2 + 2^2 + 2^3 + 2^4 + . . . + 2^n \right)\]
\[ = \frac{n\left( n + 1 \right)}{2} + 2\left( \frac{2^n - 1}{2 - 1} \right)\]
\[ = \frac{n\left( n + 1 \right)}{2} + 2^{n + 1} - 2\]