Sequences and Series - neither an AP or a GP, how do I solve it

Question

Find the sum of the first n terms of a series given $T_r = 2^r +2r - 1$

I've worked out the first six terms and found them to be; $3, 7, 13, 23,41$ and $75.$ Working out their differences we get five terms in a new series; $4, 6, 10, 18$ and $34.$ Finally working out the differences of this series we get an AP; $2, 4, 8, 16$ with the equation $T_n = 2.2^{(n-1)}$. But how do I relate this to the original series, which is neither an AP or GP, to find it's sum to $n$ numbers?

Thanks in advance!!

Answer 1

$\displaystyle\sum_{r = 1}^{n}T_r = \sum_{r = 1}^{n}(2^r+2r-1) = \sum_{r = 1}^{n}2^r + \sum_{r = 1}^{n}(2r-1)$.

This is a geometric series plus an arithmetic series.

Answer 2

$\sum_{r=1}^n T_r = \sum_{r=1}^n (2^r + 2r - 1) = \sum_{r=1}^n 2^r + 2\sum_{r=1}^n r - \sum_{r=1}^n 1 = 2^{n+1}+ n^2 - 2$. This should be the answer for general $n$.