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

Given:

$$P_n = P_1 + P_2 + \dots + P_{n-1} + 1$$

$$P_1 = 1, P_2 = 2$$

$$P_3 = 1 + 2 + 1 = 4$$

$$P_4 = 1 + 2 + 4 + 1 = 8$$

$$P_5 = 1 + 2 + 4 + 8 + 1 = 16$$

Thus, $P_1, P_2, \dots, P_8$ are 1, 2, 4, 8, 16, 32, 64, 128.

Simpler recurrence: $P_n = 2P_{n-1}, n \geq 2$ with $P_1 = 1$

Explicit formula: $P_n = 2^{n-1}$