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Show that a_n = [a_(n-1)^2/a_(n-2)] + 1 in the sequence

Source: IMO LongList 1979 - P8

May 29, 2011
algebra proposedalgebra

Problem Statement

The sequence (an)(a_n) of real numbers is defined as follows: a_1=1, \qquad a_2=2,   \text{and}   a_n=3a_{n-1}-a_{n-2} , \ \ n \geq 3. Prove that for n≥3n \geq 3, an=[an−12an−2]+1a_n=\left[ \frac{a_{n-1}^2}{a_{n-2}} \right] +1, where [x][x] denotes the integer pp such that p≤x<p+1p \leq x < p + 1.
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