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2018 AMC 12B Problem 18

Problem 18 of 25IntermediateAlgebra

A function ff is defined recursively by f(1)=f(2)=1f(1)=f(2)=1 and f(n)=f(n−1)−f(n−2)+n f(n)=f(n-1)-f(n-2)+n for all integers n≥3.n\ge3. What is f(2018)?f(2018)?

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Solution

Repeatedly substituting the recursion into itself gives f(n)=f(n−6)+6. f(n)=f(n-6)+6. So ff increases by 66 every time nn increases by 6.6. Since 2018=2+6⋅336,2018=2+6\cdot336, we have f(2018)=f(2)+6⋅336f(2018)=f(2)+6\cdot336 =1+2016=2017.=1+2016=2017. Thus, the correct answer is B.

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Concepts: recursion · telescoping

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.