If an=2an−1+1a_n = 2a_{n-1} + 1an=2an−1+1 with a0=0a_0 = 0a0=0, find Sn=∑k=0nakS_n = \sum_{k=0}^n a_kSn=∑k=0nak in closed form.
Sn=2n+1−n−2S_n = 2^{n+1} - n - 2Sn=2n+1−n−2
Sn=2n−1S_n = 2^n - 1Sn=2n−1
Sn=n⋅2n−(n+1)S_n = n \cdot 2^n - (n+1)Sn=n⋅2n−(n+1)
Sn=3(2n−1)−nS_n = 3(2^n - 1) - nSn=3(2n−1)−n