Extreme Algebra Problem

Let $P_n = a^n+b^n+c^n$ , where $n$ is a natural number. We need to find $P_5$ . Using Newton's sums or identities , then the symmetric sums $S_1 = a+b+c = 1$ , $S_2 = ab+bc+ca$ , and $S_3 = abc$ , and we have:

$\begin{aligned} P_1 & = S_1 = 1 \\ P_2 & = S_1P_1 - 2S_2 = 1-2S_2 = 2 & \small \blue{\implies S_2 = - \frac 12} \\ P_3 & = S_1P_2 - S_2P_1 + 3S_3 = 1(2)+\frac 12(1) + 3S_3 = 3 & \small \blue{\implies S_3 = \frac 16} \\ P_4 & = S_1P_3 - S_2P_2 + S_3P_1 = 1(3) + \frac 12(2) +\frac 16(1) = \frac {25}6 \\ P_5 & = S_1P_4 - S_2P_3 + S_3P_2 = \frac {25}6 + \frac 32 +\frac 26 = \boxed 6 \end{aligned}$