Question

A function $f$ is defined recursively by $f(1) = f(2) = 1$ and

$f(n) = f(n - 1) - f(n - 2) + n$

for all integers $n \ge 3.$ What is $f(2018)?$

$2016$

$2017$

$2018$

$2019$

$2020$

Answer: B

Concepts:recursion pattern recognition

Difficulty rating: 1910

Small Hint:

Look for a particular solution of the form $f(n) = n + c;$ subtracting it leaves $g(n) = g(n-1) - g(n-2)$

Big Hint:

That homogeneous recursion is periodic with period $6;$ reduce $2018$ modulo $6$

Solution:

Notice $f(n) = n + 1$ solves the recurrence on its own, so write $f(n) = (n + 1) + g(n).$ Then $g$ satisfies the homogeneous version $g(n) = g(n-1) - g(n-2).$ With $g(1) = -1$ and $g(2) = -2,$ it cycles with period $6$ : $-1, -2, -1, 1, 2, 1, \ldots.$ Since $2018 \equiv 2 \pmod 6,$ we get $g(2018) = -2,$ so $f(2018) = 2019 - 2 = 2017.$ Therefore, the answer is B.

2018 AMC 10B Problem 20

Answer: B

Difficulty rating: 1910

Small Hint:

Big Hint:

Solution:

Problem 20 in Other Years