limit of a sequence $u_n = (3+\sqrt{5} )^n -(3-\sqrt{5} )^n$.

Question

I would like to know the limit of the sequence: $u_n = (3+\sqrt{5} )^n -(3-\sqrt{5} )^n$.

I tried to involve the binomial theorem but didn't find the answer

Answer 1

$a>1>b>0 \implies a^n-b^n \to \infty$ as $n \to \infty$ ,since $b^n \to 0$

Answer 2

We have $$ u_n = \underbrace{a^n}_{\to\infty} - \underbrace{b^n}_{\to 0} $$ with $a > 1$ and $0 < b < 1$, so $$ \lim_{n\to\infty} u_n = \infty $$