HMMT February 2015 #0jna

HMMT February 2015Hard

From module Telescoping Sums and Products

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Recurrence relationsTelescoping series

Problem

Problem:

The sequences of real numbers $\{a_{i}\}_{i=1}^{\infty}$ and $\{b_{i}\}_{i=1}^{\infty}$ satisfy $a_{n+1} = (a_{n-1} - 1)(b_{n} + 1)$ and $b_{n+1} = a_{n} b_{n-1} - 1$ for $n \geq 2$ , with $a_{1} = a_{2} = 2015$ and $b_{1} = b_{2} = 2013$ . Evaluate, with proof, the infinite sum

\sum_{n=1}^{\infty} b_{n}\left(\frac{1}{a_{n+1}}-\frac{1}{a_{n+3}}\right)

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