An algebra problem by Shevy Doc

This type of problems can be solved using Newton's sums (identities) as follows. Let $P_n = a^n+b^n +c^n$ , where $n$ is a positive integer; $S_1 = a+b+c=1$ , $S_2 = ab+bc+ca$ , and $S_3 = abc$ . Then we have:

$\begin{aligned} P_1 & = S_1 =1 \\ P_2 & = S_1P_1 - 2S_2 = 1 - 2S_2 = 2 & \small \implies S_2 = - \frac 12 \\ P_3 & = S_1P_2 - S_2 P_1 + 3S_3 = 1 \times 2 + \frac 12 \times 1 + 3S_3 = 3 & \small \implies S_3 = \frac 16 \\ P_4 & = S_1 P_3 - S_2 P_2 + S_3P_1 = 1 \times 3+\frac 12 \times 2 + \frac 16 \times 1 = \frac {25}6 \\ P_5 & = S_1 P_4 - S_2 P_3 + S_3P_2 = 1 \times \frac {25}6 +\frac 12 \times 3 + \frac 16 \times 2 = \boxed 6 \end{aligned}$