2. Given real numbers $a, b, x, y$ satisfy the system of equations:
$$
\left\{\begin{array}{l}
a x+b y=3, \\
a x^{2}+b y^{2}=7, \\
a x^{3}+b y^{3}=16, \\
a x^{4}+b y^{4}=42 .
\end{array}\right.
$$

Find the value of $a x^{5}+b y^{5}$.
提示 Set
$$
c=x+y, d=-x y, a_{n}=a x^{n}+b y^{n} \text {. }
$$

then $a_{n+1}=c a_{n}+d a_{n-1}$.
$$
\text { Hence }\left\{\begin{array}{l}
7 c+3 d=16, \\
16 c+7 d=42, \\
42 c+16 d=a_{5}
\end{array}\right. \text {. }
$$

Solving yields $a_{5}=20$.