How to find n-element of a linear recurrent sequence

I am trying to find 1000-element of these linear recurrent sequences:

$1.$ $x_{i+5}=2x_{i+4}+5x_{i+3}+3x_{i+2}+4x_i$ if $x_{0,1,2,3,4}=(0,1,2,3,4)$. And we work over $GF(7)$.

$2.$ $y_{i+2}=2y_{i+1}-y_i+1$ if $y_{0,1} = (0,1)$. And we work over the field of complex numbers $C$.

Could you describe the approaches to solving these problems? Please write in detail!

• Have you tried computing a few terms and looking for a pattern (and then proving the pattern continues)? – Gerry Myerson May 14 '18 at 9:45
• Good advice from #Gerry Myerson. (1) repeats with a period of 21 terms and (2) has an obvious pattern that suggests a simple closed form solution – gandalf61 May 14 '18 at 12:12
• @gandalf61 It would be very cool if you write it in detail, please. I don't really know how to prove this periodicity – alexhak May 14 '18 at 14:18
• @GerryMyerson It would be very cool if you write it in detail, please. I don't really know how to prove this periodicity – alexhak May 14 '18 at 14:19
• It would be very cool if you, alexhak, would compute the terms and see the pattern first. – Gerry Myerson May 14 '18 at 21:27