HMMT February

Algebra Difficulty 5.1 AIME, harder Find the answer

An infinite sequence of positive real numbers is defined by $a_{0}=1$ and $a_{n+2}=6 a_{n}-a_{n+1}$ for $n=0,1,2, \cdots$ Find the possible value(s) of $a_{2007}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The characteristic equation of the linear homogeneous equation is $m^{2}+m-6=(m+3)(m-2)=0$ with solutions $m=-3$ and $m=2$ . Hence the general solution is given by $a_{n}=A(2)^{n}+B(-3)^{n}$ where $A$ and $B$ are constants to be determined. Then we have $a_{n}>0$ for $n \geq 0$ , so necessarily $B=0$ , and $a_{0}=1 \Rightarrow A=1$ . Therefore, the unique solution to the recurrence is $a_{n}=2^{n}$ for all $n$ .

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