Suppose \(P_1 = 1,\, P_2 = 2\) and for \(n>2,\,P_n = P_1 + P_2 + ... + P_{n−1} + 1\). Find the values of \(P_1, P_2, ..., P_8\). Can you find a simpler recursive formula for \(P_n\)? Can you give an explicit formula?

\(P_1 = 1,\, P_2 = 2\)
\(P_n = P_1 + P_2 + ... + P_{n−1} + 1\)

\(P_3 = 1 + 2 + 1 = 4\)
\(P_4 = 1 + 2 + 4 + 1 = 8\)
\(P_5 = 1 + 2 + 4 + 8 + 1 = 16\)
\(P_6 = 1 + 2 + 4 + 8 + 16 + 1 = 32\)
\(P_7 = 1 + 2 + 4 + 8 + 16 + 32 + 1 = 64\)
\(P_8 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 1 = 128\)
The values are: 1, 2, 4, 8, 16, 32, 64, 128

\(P_n = 2P_{n−1} \) for \(n ≥ 2\)

\(P_n = 2^{n−1}\)