\((3, 5, 9, 17, 33, \ldots )\text{.}\) Each term in this sequence is one more than a power of 2, so we might guess the closed formula is \(c_n = a_n+1 = 2^n + 1\text{.}\) If we try this though, we get \(c_0 = 2^0 + 1 = 2\) and \(c_1 = 2^1 + 1 = 3\text{.}\) We are off because the indices are shifted. What we really want is \(c_n = a_{n+1}+1\) giving \(c_n = 2^{n+1} + 1\text{.}\)
in-context