Competition problem — Algebra — Maths Olympiad Prep

Algebra Difficulty 4.8 AIME Prove it Brazil

The sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = 8$ , $a_2 = 18$ , $a_{n+2} = a_{n+1} \cdot a_n$ .
Find all terms which are perfect squares.

Solution

We have $a_n = 2^{b_n} 3^{c_n}$ , where $b_{n+2} = b_n + b_{n+1}$ , and $c_{n+2} = c_n + c_{n+1}$ .

$c_1$ and $c_2$ are even, so all $c_n$ are even.

$b_1$ and $b_2$ are odd, so $b_n$ is even for $n$ multiple of $3$ .

We need both $b_n$ and $c_n$ even for $a_n$ to be a perfect square.

So the answer is all $n$ multiple of $3$ .

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