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a_{n+3}=7a_{n+2}-11a_{n+1}+5a_n-3\cdot2^n

Source: Moldova EGMO TST 2019

July 10, 2023
Sequence

Problem Statement

The sequence (an)n≥1(a_n)_{n\geq1} is defined as: a1=2,a2=20,a3=56,an+3=7an+2−11an+1+5an−3⋅2n.a_1=2, a_2=20, a_3=56, a_{n+3}=7a_{n+2}-11a_{n+1}+5a_n-3\cdot2^n. Prove that ana_n is positive for every positive integer nn{}. Find the remainder of the divison of a673a_{673} to 673673.
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