solve recurrence relation $a_{n+2}=(a_{n+1}+1)/a_n$

Question

Let $a_1$ = 2015, $a_2$=2016. Find $a_{2017}$, given $a_{n+2}=(a_{n+1}+1)/a_n$ I have found the 2 roots of the characteristic equation $x^2-x-1=0$ : $r_1 = ((1+\sqrt{5})/2)^n$ , $r_2 = ((1-\sqrt{5})/2)^n$ Then I have used a calculator to find that the coefficients a and b are -57.418 and 56.7999 respectively. However every calculator treats raising to the power of 2017 as an error. Where did I go wrong?

Answer 1

Note $a_{1} = 2015, a_{2} = 2016, a_{3} = \frac{2017}{2015}, a_{4} = \frac{2}{2015},a_{5} = 1, a_{6} = 2015, a_{7} = 2016,...$

Thus the sequence repeats modulo $5$. Hence $a_{2017} = a_{2} = 2016$.

You cannot use characteristic equations (as pointed out by other people such as jjagmath).

Answer 2

Calling $a_1 = c_1$ and $a_2 = c_2$ we have

$$ \cases{ a_1 = c_1\\ a_2 = c_2\\ a_3 = \frac{c_2+1}{c_1}\\ a_4 = \frac{c_1+c_2+1}{c_1c_2}\\ a_5 = \frac{c_1+1}{c_2}\\ a_6 = c_1\\ a_7 = c_2\\ \vdots } $$