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upper bound of 2008th term

Source: ARO 2008, 10th Grade P4

June 13, 2008
limitalgebra

Problem Statement

The sequences (an),(bn) (a_n),(b_n) are defined by a_1\equal{}1,b_1\equal{}2 and a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n},   b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.
Show that a2008<5 a_{2008} < 5.
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