Let $a_1=b_1=1$ and define sequences $a_n$ and $b_n$ recursively by
Thus
and so on.
Find an explicit formula for $a_n$ and for $b_n$ in terms of $n$.
Look at the sequence = {1, 0, -2, -4, -4, 0, 8, 16, 16, 0, -32, -64,.....}
we notice that it comes in blocks of four numbers each being (-4) times the corresponding number in the previous block
this means
Proof:
so that's one recurrence relation with one variable with four initial conditions
solution:
similarly