3. Real numbers $a, b, x$ and $y$ satisfy the system of equations
$$
\left\{\begin{array}{l}
a x+b y=3, \\
a x^{2}+b y^{2}=7, \\
a x^{3}+b y^{3}=6, \\
a x^{4}+b y^{4}=42 .
\end{array}\right.
$$

Find the value of $a x^{5}+b y^{5}$.
3. Let $T_{n}=a x^{n}+b y^{n}$, where $x, y$ are the two real roots of the quadratic equation $t^{2}=p t+q$. According to Newton's formula, we have $T_{n}=p T_{n-1}+q T_{n-2}$, thus $\left\{\begin{array}{l}T_{3}=p T_{2}+q T_{1} \\ T_{4}=p_{3}+q T_{2}\end{array}\right.$, which means $\left\{\begin{array}{l}16=7 p+3 q \\ 42=16 p+7 q\end{array} \Rightarrow\left\{\begin{array}{l}p=-14, \\ q=38 .\end{array}\right.\right.$ Therefore, $T_{5}=p T_{4}+q T_{3}=-14 \times 42+38 \times 16=20$, which means $a x^{5}+b y^{5}=20$.