Example 4 Real numbers $a, b, x, y$ satisfy $a x + b y = 3$, $a x^2 + b y^2 = 7$, $a x^3 + b y^3 = 16$, $a x^4 + b y^4 = 42$. Find the value of $a x^5 + b y^5$.
(8th American High School Mathematics Examination)
Explanation: Let $S_{n}=a x^{n}+b y^{n}$. Then
$$
S_{n}=(x+y) S_{n-1}-x y S_{n-2}, n=3,4, \cdots \text {. }
$$

Substituting the known conditions into the above equation, we get
$$
\begin{array}{l}
7(x+y)-3 x y=16,16(x+y)-7 x y=42 . \\
\text { Solving, we get } x+y=-14, x y=-38 . \\
\text { Therefore, } S_{n}=-14 S_{n-1}+38 S_{n-2} \quad(n \geqslant 3) . \\
\text { Hence, } S_{5}=a x^{5}+b y^{5}=-14 S_{4}+38 S_{3} \\
=-14 \times 42+38 \times 16=20 .
\end{array}
$$