Limit of $13^{a_n}=12^{a_{n-1}}+5^{a_{n-2}}$

Question

I have the following recurrence: $$13^{a_n}=12^{a_{n-1}}+5^{a_{n-2}},\;a_0=0,\;a_1=1.$$ I have to prove that it is monotone and bounded, and I have to find the limit as $n\to\infty$. It was for an older competition, but I cannot find an answer to it. I tried reformulating it to $$13^{a_n}=13^{a_{n-1}\alpha}+13^{a_{n-2}\beta},\;\alpha=\log_{13}12,\;\beta=\log_{13}5,$$ then with $b_n=13^{a_n}$ it is $$b_n=b_{n-1}^{\alpha}+b_{n-2}^{\beta}.$$ If it is the right way, how to proceed, if not, can you give me a hint how to start? I never looked deep into recurrence relations, but this is for first year engineering students, so advanced techniques are not needed I think.

Answer 1

$(5,12,13)$ is a Pythagorean triple. $13^2 = 12^2 + 5^2$

Use induction to show that if $0<a_{n-2}<a_{n-1}<2$ then $a_{n-1}<a_n<2$