Question 93
The sequences $\left(a_{n}\right)_{n=0}^{\infty}$ and $\left(b_{n}\right)_{n=0}^{\infty}$ are defined by the recurrence relations
$$
a_{0}=0, \quad a_{1}=1, \quad a_{n+1}=\frac{2018}{n} a_{n}+a_{n-1} \quad \text{ for } n \geqslant 1
$$
and
$$
b_{0}=0, \quad b_{1}=1, \quad b_{n+1}=\frac{2020}{n} b_{n}+b_{n-1} \quad \text{ for } n \geqslant 1
$$
Prove:
$$
\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}
$$
$$
a_{0}=0, \quad a_{1}=1, \quad a_{n+1}=\frac{2018}{n} a_{n}+a_{n-1} \quad \text{ for } n \geqslant 1
$$
and
$$
b_{0}=0, \quad b_{1}=1, \quad b_{n+1}=\frac{2020}{n} b_{n}+b_{n-1} \quad \text{ for } n \geqslant 1
$$
Prove:
$$
\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}
$$