Given Sn=∑k=0nakS_n = \sum_{k=0}^n a_kSn=∑k=0nak where Sn=2Sn−1+1S_n = 2S_{n-1} + 1Sn=2Sn−1+1 with S0=1S_0 = 1S0=1, find a closed form for ana_nan when n≥1n \geq 1n≥1.
an=2n−1a_n = 2^n - 1an=2n−1
an=2na_n = 2^nan=2n
an=2n−1a_n = 2^{n-1}an=2n−1
an=2n+1−2a_n = 2^{n+1} - 2an=2n+1−2