Suppose the real numbers a,b,x, and y satisfy the equations

Suppose the real numbers a,b,x, and y satisfy the equations

     ax+by=3,

     ax^2+by^2=7,

     ax^3+by^3=16,

     ax^4+by^4=42.  

Evaluate ax^5+by^5.

$\begin{array}{|rcll|} \hline && ax^5 + b y^5 \\ &=& ( \frac{49}{76}-\frac{457}{76 \cdot \sqrt{87}}) \cdot (-7-\sqrt{87})^5 +( \frac{49}{76}+\frac{457}{76 \cdot\sqrt{87}}) \cdot (\sqrt{87}-7)^5 \\ &=& 20 \\ \hline \end{array}$

Here's an alternative approach (no need to find a, b x and y explicitly):

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Very crafty , Alan !!!!!......I was anxious to see how this  could be solved without having to resort to CAS input