Write the sum to n terms of a series whose rth term is

series whose rth term is r + 2r.

$\left(1+2^{1}\right)+\left(2+2^{2}\right)+\left(3+2^{3}\right)+\left(4+2^{4}\right)+\ldots+\left(n+2^{n}\right)$

Thus, we have:

$S_{n}=\left(1+2^{1}\right)+\left(2+2^{2}\right)+\left(3+2^{3}\right)+\left(4+2^{4}\right)+\ldots+\left(n+2^{n}\right)$

$=(1+2+3+4+\ldots+n)+\left(2+2^{2}+2^{3}+2^{4}+\ldots+2^{n}\right)$

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

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