Convergent sequence involving induction?

Question

Let a sequence be defined by x(n+1) = ( x(n)^5 +2)/6 with x(1)=0.

Show x(n+1)-x(n) =< 5/6 [x(n)-x(n-1)] =< 1/3 (5/6)^(n-1)

Presumably we can solve this using induction, though it seems quite tricky and I've made a mess of it so far. Or is there a better method? Thanks very much.

Answer 1

This is fairly straightforward:
We have
x(n+1)=(x(n)^5 +2)/6 and
x(n)=(x(n-1)^5 +2)/6, so
x(n+1)-x(n)=(1/6)[x(n)^5 -x(n-1)^5].
Now factor x(n)^5-x(n-1)^5 and use the fact that x(n) is always between 0 and 1 (which can also be shown by induction).

Answer 2

See http://answers.yahoo.com/question/?qid=20071117155243AARKxCQ