Recursive sequence satisfies inequality
Source: IMO Shortlist 1996, A3
August 9, 2008
inequalitiesalgebraSequenceRecurrenceIMO Shortlist
Problem Statement
Let be given, and starting a_0 \equal{} 1, a_1 \equal{} a define recursively:
a_{n\plus{}1} \equal{} \left(\frac{a^2_n}{a^2_{n\minus{}1}} \minus{} 2 \right) \cdot a_n.
Show that for all integers we have: \sum^k_{i \equal{} 0} \frac{1}{a_i} < \frac12 \cdot (2 \plus{} a \minus{} \sqrt{a^2\minus{}4}).