jmc

Question

A sequence of integers is defined as follows: a_i = i for 1 i 5, and a_i = a_1 a_2 a_i - 1 - 1for i > 5. Evaluate a_1 a_2 a_2011 - _i = 1^2011 a_i^2.

Accepted Answer

For i 6, a_i = a_1 a_2 a_i - 1 - 1. So align* a_i + 1 &= a_1 a_2 a_i - 1 &= (a_1 a_2 a_i - 1) a_i - 1 &= (a_i + 1) a_i - 1 &= a_i^2 + a_i - 1. align*Then a_i^2 = a_i + 1 - a_i + 1, so align* a_1 a_2 a_2011 - _i = 1^2011 a_i^2 &= a_2012 + 1 - (a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2) - _i = 6^2011 (a_i + 1 - a_i + 1) &= a_2012 + 1 - (a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2) - (a_2012 - a_6 + 2006) &= a_6 - (a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2) - 2005 &= 119 - (1^2 + 2^2 + 3^2 + 4^2 + 5^2) - 2005 &= -1941. align*