how to solve these equation?

Question

For $a , b , x , y$ are members of $\mathbb{R}$

If

$ax+by=3\\ax^2+by^2=7\\ax^3+by^3=16\\ax^4+by^4=42$

then
$ax^5+by^5=?$

a lot of thanks for all comments

See this: http://www.artofproblemsolving.com/Wiki/index.php/1990_AIME_Problems/Problem_15 — JimmyK4542, Jul 17 '14 at 21:20
Related: http://math.stackexchange.com/questions/6965/system-of-equations — Aryabhata, Jul 18 '14 at 06:26
Current link: https://artofproblemsolving.com/wiki/index.php?title=1990_AIME_Problems/Problem_15 (previous link is broken) — bluemaster, Sep 12 '17 at 22:38

score 1 · Answer 1 · answered Jul 18 '14 at 06:32

Let $x,y$ be roots of $z^2 + uz + v = 0$

Mutiplying the first by $v$ and second by $u$ and adding to the third gives

$$16 + 7u + 3v = 0 $$

Similary we get

$$ 42 + 16u + 7v = 0 $$

This allows you to solve for $u$ and $v$.

If $ax^5 + by^5 = X$ then we get in a similar fashion that

$$ X + 42u + 16v = 0$$

I leave the rest to you.

1 Answers1