# Find the general formula of $u_n$:

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

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What sort of class is this for? –  andybenji Dec 17 '12 at 18:08

## 2 Answers

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 $n\geq 1$.

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

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

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