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Philippine Mathematical Olympiad

Algebra Difficulty 5.3 AIME, harder Prove it Philippines

Problem:
Let $\{a_n\}$ be a sequence such that $a_1 = 20$ , $a_2 = 17$ and $a_n + 2 a_{n-2} = 3 a_{n-1}$ . Determine the value of $a_{2017} - a_{2016}$ .

Solution

Solution:
We have $a_n = 3 a_{n-1} - 2 a_{n-2} \Longrightarrow a_n - a_{n-1} = 2(a_{n-1} - a_{n-2})$ . Repeated use of this recurrence relation gives $a_{2017} - a_{2016} = 2^{2015}(a_2 - a_1) = -3 \cdot 2^{2015}$ .

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