<s>[INST] Determine the value of $ax^5 + by^5$ given that $a$, $b$, $x$, and $y$ satisfy the following system: Given

$\begin{cases} ax + by = 4 \\ ax^2 + by^2 = 8 \\ ax^3 + by^3 = 17 \\ ax^4 + by^4 = 42 \end{cases} $

Determine the value of $ax^5 + by^5$
Do you have any brilliant idea to solve this problem? By looking at the right-hand side, it is actually a sequence of the sum of quadratic numbers. But, of course it is so irrational to say that $ax^5 + by^5 = 42 + 36 = 78$ just by looking at that point. 
 [/INST] You have
$$ (x+y)(ax^2+by^2) = ax^3+by^3+xy(ax+by) \implies 8(x+y) = 17+4xy  $$
$$ (x+y)(ax^3+by^3) = ax^4+by^4 + xy(ax^2+by^2) \implies 17(x+y) = 42 + 8xy $$
It follows that 
$$ x+y = 8 $$
$$ xy = \frac{47}{4} $$
Also, from the same relations
$$ ax^5 + by^5 = (x+y)(ax^4+by^4) - xy(ax^3+by^3) = 42(x+y)-17xy = \frac{545}{4} $$
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