HMMT 2009 A8

If $a, b, x$, and $y$ are real numbers such that $a x+b y=3, a x^{2}+b y^{2}=7, a x^{3}+b y^{3}=16$, and $a x^{4}+b y^{4}=42$, find $a x^{5}+b y^{5}$.