Find the general formula of $u_n$:

Question

Find the general formula of $u_n$: $$\begin{cases}u_1=\frac{5}{4}\\[10pt]u_{n+1}=8u_n^4-8u_n^2+1, \forall n \in \mathbb{N} \end{cases}$$

What sort of class is this for? – Andrew Maurer Dec 17 '12 at 18:08 — Andrew Maurer, Dec 17 '12 at 18:08

score 0 · Answer 1 · answered Dec 17 '12 at 16:40

0

Wolfram gives $u_n = \cosh{(4^{n-1}\log2)}$.

answered Dec 17 '12 at 16:40

dexter04

1,971

froggie · Accepted Answer · 2012-12-17T18:56:58.490

0

The trick here is to realize that your polynomial is a Chebyshev Polynomial. These polynomials have nice properties you can exploit.

Define $P(x) = x^4$, $L(x) = \frac{1}{2}(x + x^{-1})$, and $Q(x) = 8x^4 - 8x^2 + 1$. You can verify directly that $$L\circ P = Q\circ L.$$ Note, moreover, that $5/4 = L(2)$. Then $$u_n = Q^{n-1}(5/4) = Q^{n-1}(L(2)) = L(P^{n-1}(2)) = L(2^{4^{n-1}}) = \frac{1}{2}\left(2^{4^{n-1}} + 2^{-4^{n-1}}\right)$$ for each $n\geq 1$.

edited Dec 17 '12 at 18:56

answered Dec 17 '12 at 18:51

froggie

9,941

Note also that this agrees with what @dexter04 found with Wolfram. – froggie Dec 17 '12 at 18:56

Find the general formula of $u_n$:

2 Answers2