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Difference between revisions of "2018 AMC 10B Problems/Problem 20"

Problem

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 $n \geq 3$. What is $f(2018)$? $\textbf{(A)} \text{ 2016} \qquad \textbf{(B)} \text{ 2017} \qquad \textbf{(C)} \text{ 2018} \qquad \textbf{(D)} \text{ 2019} \qquad \textbf{(E)} \text{ 2020}$

Solution 1 $f\left(n\right) = f\left(n - 1\right) - f\left(n - 2\right) + n$ $= \left(f\left(n - 2\right) - f\left(n - 3\right) + n - 1\right) - f\left(n - 2\right) + n$ $= 2n - 1 - f\left(n - 3\right)$ $= 2n - 1 - \left(2\left(n - 3\right) - 1 - f\left(n - 6\right)\right)$ $= f\left(n - 6\right) + 6$

Thus, $f\left(2018\right) = 2016 + f\left(2\right) = 2017$. $\boxed{B}$

Solution 2 (A Bit Bashy)

Start out by listing some terms of the sequence. $$f(1)=1$$ $$f(2)=1$$ $$f(3)=3$$ $$f(4)=6$$ $$f(5)=8$$ $$f(6)=8$$ $$f(7)=7$$ $$f(8)=7$$ $$f(9)=9$$ $$f(10)=12$$ $$f(11)=14$$ $$f(12)=14$$ $$f(13)=13$$ $$f(14)=13$$ $$f(15)=15$$ $$.....$$ Notice that $f(n)=n$ whenever $n$ is an odd multiple of $3$, and the pattern of numbers that follow will always be +3, +2, +0, -1, +0. The closest odd multiple of $3$ to $2018$ is $2013$, so we have $$f(2013)=2013$$ $$f(2014)=2016$$ $$f(2015)=2018$$ $$f(2016)=2018$$ $$f(2017)=2017$$ $$f(2018)=\boxed{(B) 2017}.$$

Solution 3(Bashy Pattern Finding)

Writing out the first few values, we get: $1,1,3,6,8,8,7,7,9,12,14,14,13,13,15,18,20,20,19,19...$. Examining, we see that every number $x$ where $x \equiv 1 \pmod 6$ has $f(x)=x$, $f(x+1)=f(x)=x$, and $f(x-1)=f(x-2)=x+1$. The greatest number that's 1 (mod 6) and less $2018$ is $2017$, so we have $f(2017)=f(2018)=2017.$ \boxed B\$ (Random_Guy)

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