Competition problem — Algebra — Maths Olympiad Prep

Library / /102 of 860

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. Spacing and $ signs are ignored.

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.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.