Question 175
a) Prove that for each positive integer $n$ there is a unique positive integer $a_{n}$ such that
$$
(1+\sqrt{5})^{n}=\sqrt{a_{n}}+\sqrt{a_{n}+4^{n}}
$$
b) Prove that $a_{2010}$ is divisible by $5 \cdot 4^{2009}$ and find the quotient.
$$
(1+\sqrt{5})^{n}=\sqrt{a_{n}}+\sqrt{a_{n}+4^{n}}
$$
b) Prove that $a_{2010}$ is divisible by $5 \cdot 4^{2009}$ and find the quotient.