Problem 6

A sequence

(u_n)

is defined by

u_0=2

u_1=\frac52

, and

u_{n+1}=u_n(u_{n-1}^2-2)-u_1

for

n=1,2,\ldots

. Prove that for every positive integer

n

\lfloor u_n\rfloor=2^{\frac{2^n-(-1)^n}{3}}

, where

\lfloor x\rfloor

denotes the greatest integer less than or equal to

x

Step 3 of 6: Build the model sequence

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In plain words

This is a “change of coordinates”: the nonlinear-looking recurrence for $u_n$ becomes linear addition in the exponent $x_n$ .

a_n=2^{x_n}+2^{-x_n}

Detailed analysis

Let $a_n=2^{x_n}+2^{-x_n}$ . The initial values match: $a_0=2=u_0$ and $a_1=2+\frac12=\frac52=u_1$ . We will show that $a_n$ obeys the same recurrence as $u_n$ .