<s>[INST] Systems of polynomial equations involving sums of equal powers Given the following system of polynomial equations:
$$
\left\{\begin{array}{lclclcr}
x & + & y & + & z & = & 1
\\
x^{2} & + & y^{2} & + & z^{2} & = & 14
\\
x^{3} & + & y^{3} & + & z^{3} & = & 36
\end{array}\right.
$$
What is $x^{5} + y^{5} + z^{5}\ {\large ?}$ .

How should I approach this? Is there a general formula for this kind of system?
 [/INST] The requested exponent does not matter that much. Each of the three is a root of the same
$$  6 t^3 - 6 t^2  - 39 t - 31, $$
one real and two complex conjugates, but we don't need them. Each also obeys
$$  6 t^{n+3} - 6 t^{n+2}  - 39 t^{n+1} - 31 t^n, $$ so that $a_n=x^n + y^n + z^n$  is a sequence with linear recurrence
$$ a_{n+3} = a_{n+2} + \frac{13}{2} a_{n+1} + \frac{31}{6} a_n  $$ or
$$ a_1 = 1, \; a_2 = 14, \; a_3 = 36, \; a_4 = \frac{793}{6}, \; $$
$$ a_5 = \frac{877}{2}, \; a_6 = \frac{17803}{12}, \; a_7 = \frac{180601}{36}, \; a_8 = \frac{1218641}{72}, \; $$
</s>