HMMT February

Algebra Difficulty 4.8 AIME Find the answer

Let $a_{0}, a_{1}, a_{2}, \ldots$ denote the sequence of real numbers such that $a_{0}=2$ and $a_{n+1}=\frac{a_{n}}{1+a_{n}}$ for $n \geq 0$ . Compute $a_{2012}$ .

Your answer

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

Solution

Calculating out the first few terms, note that they follow the pattern $a_{n}=\frac{2}{2 n+1}$ . Plugging this back into the recursion shows that it indeed works.

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