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

Revision as of 16:18, 23 July 2023

The following problem is from both the 2018 AMC 12B #18 and 2018 AMC 10B #20, so both problems redirect to this page.

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 $f(2018)$ ?

$\textbf{(A) } 2016 \qquad \textbf{(B) } 2017 \qquad \textbf{(C) } 2018 \qquad \textbf{(D) } 2019 \qquad \textbf{(E) } 2020$

Solution 1 (Algebra)

For all integers $n \geq 7,$ note that $\begin{align*} f(n)&=f(n-1)-f(n-2)+n \\ &=[f(n-2)-f(n-3)+n-1]-f(n-2)+n \\ &=-f(n-3)+2n-1 \\ &=-[f(n-4)-f(n-5)+n-3]+2n-1 \\ &=-f(n-4)+f(n-5)+n+2 \\ &=-[f(n-5)-f(n-6)+n-4]+f(n-5)+n+2 \\ &=f(n-6)+6. \end{align*}$ It follows that $\begin{align*} f(2018)&=f(2012)+6 \\ &=f(2006)+12 \\ &=f(2000)+18 \\ & \ \vdots \\ &=f(2)+2016 \\ &=\boxed{\textbf{(B) } 2017}. \end{align*}$ ~MRENTHUSIASM

Solution 2 (Algebra)

For all integers $n\geq3,$ we rearrange the given equation: $f(n)-f(n-1)+f(n-2)=n. \hspace{28.25mm}(1)$ For all integers $n\geq4,$ it follows that $f(n-1)-f(n-2)+f(n-3)=n-1. \hspace{15mm}(2)$ For all integers $n\geq4,$ we add $(1)$ and $(2):$ $f(n)+f(n-3)=2n-1. \hspace{38.625mm}(3)$ For all integers $n\geq7,$ it follows that $f(n-3)+f(n-6)=2n-7. \hspace{32mm}(4)$ For all integers $n\geq7,$ we subtract $(4)$ from $(3):$ $f(n)-f(n-6)=6. \hspace{47.5mm}(5)$ From $(5),$ we have the following system of $336$ equations: $\begin{align*} f(2018)-f(2012)&=6, \\ f(2012)-f(2006)&=6, \\ f(2006)-f(2000)&=6, \\ & \ \vdots \\ f(8)-f(2)&=6. \end{align*}$ We add these equations up to get $f(2018)-f(2)=6\cdot336=2016,$ from which $f(2018)=f(2)+2016=\boxed{\textbf{(B) } 2017}.$

~AopsUser101 ~MRENTHUSIASM

Solution 3 (Finite Differences)

Preamble: In this solution, we define the sequence $A$ to satisfy $a_n = f(n),$ where $a_n$ represents the $n$ th term of the sequence $A.$ This solution will show a few different perspectives. Even though it may not be as quick as some of the solutions above, I feel like it is an interesting concept, and may be more motivated.

To begin, we consider the sequence $B$ formed when we take the difference of consecutive terms between $A.$ Define $b_n = a_{n+1} - a_n.$ Notice that for $n \ge 4,$ we have

$\begin{cases}\begin{aligned} a_{n+1} &= a_{n} - a_{n-1} + (n+1) \\ a_{n} &= a_{n-1} - a_{n-2} + n \end{aligned}.\end{cases}$

Notice that subtracting the second equation from the first, we see that $b_{n} = b_{n-1} - b_{n-2} + 1.$

If you didn’t notice that $B$ repeated directly in the solution above, you could also, possibly more naturally, take the finite differences of the sequence $b_n$ so that you could define $c_n = b_{n+1} - b_n.$ Using a similar method as above through reindexing and then subtracting, you could find that $c_n = c_{n-1} - c_{n-2}.$ The sum of any six consecutive terms of a sequence which satisfies such a recursion is $0,$ in which you have that $b_{n} = b_{n+6}.$ In the case in which finite differences didn’t reduce to such a special recursion, you could still find the first few terms of $C$ to see if there are any patterns, now that you have a much simpler sequence. Doing so in this case, it can also be seen by seeing that the sequence $C$ looks like $\underbrace{2, 1, -1, -2, -1, 1,}_{\text{cycle period}} 2, 1, -1, -2, -1, 1, \ldots$ in which the same result follows.

Using the fact that $B$ repeats every six terms, this motivates us to look at the sequence $B$ more carefully. Doing so, we see that $B$ looks like $\underbrace{2, 3, 2, 0, -1, 0,}_{\text{cycle period}} 2, 3, 2, 0, -1, 0, \ldots$ (If you tried pattern finding on sequence $B$ directly, you could also arrive at this result, although I figured defining a second sequence based on finite differences was more motivated.)

Now, there are two ways to finish.

Finish Method #1: Notice that any six consecutive terms of $B$ sum to $6,$ after which we see that $a_n = a_{n-6} + 6.$ Therefore, $a_{2018} = a_{2012} + 6 = \cdots = a_{2} + 2016 = \boxed{\textbf{(B) } 2017}.$

Finish Method #2: Notice that $a_{2018} = a_{2017} - a_{2016} + 2018 = B_{2016} + 2018 = -1 + 2018 = \boxed{\textbf{(B) } 2017}.$

~Professor-Mom

Solution 4 (Pattern)

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

Solution 5 (Pattern)

Writing out the first few values, we get $1,1,3,6,8,8,7,7,9,12,14,14,13,13,15,18,20,20,19,19,\ldots.$ 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 $1\pmod{6}$ and less than $2018$ is $2017,$ so we have $f(2017)=f(2018)=\boxed{\textbf{(B) } 2017}.$

Solution 6

$f(n)=f(n-1)-f(n-2)+n$ $\newline$ $f(n-1)=f(n-2)-f(n-3)+n-1$ $\newline$ Subtracting those two and rearranging gives $f(n)-2f(n-1)+2f(n-2)-f(n-3)=1$ $\newline$ $f(n-1)-2f(n-2)+2f(n-3)-f(n-4)=1$ $\newline$ Subtracting those two gives $f(n)-3f(n-1)+4f(n-2)-3f(n-3)+f(n-4)=0$ $\newline$ The characteristic polynomial is $x^4-3x^3+4x^2-3x+1=0$ $\newline$ $x=1$ is a root, so using synthetic division results in $(x-1)(x^3-2x^2+2x-1)=0$ $\newline$ $x=1$ is a root, so using synthetic division results in $(x-1)^2(x^2-x+1)=0$ $\newline$ $x^2-x+1=0$ has roots $x=\frac{1}{2}\pm\frac{i\sqrt{3}}{2}$ $\newline$ And $f(n)=(An+D)*1^n+B*(\frac{1}{2}-\frac{i\sqrt{3}}{2})^n+C*(\frac{1}{2}+\frac{i\sqrt{3}}{2})^n$ $\newline$ Plugging in $n=1$ , $n=2$ , $n=3$ , and $n=4$ results in a system of $3$ linear equations $\newline$ Solving them gives $A=1, B=\frac{1}{2}-\frac{i\sqrt{3}}{2}, C=\frac{1}{2}+\frac{i\sqrt{3}}{2}, D=1$ (note: you can guess A=1 by answer choices) $\newline$ So plugging in $n=2018$ results in $2018+1+(\frac{1}{2}-\frac{i\sqrt{3}}{2})^{2019}+(\frac{1}{2}+\frac{i\sqrt{3}}{2})^{2019}=2019+(cos(-60)+sin(-60))^{2019})+(cos(60)+sin(60))^{2019})=2019+(cos(-60*2019)+sin(-60*2019))+(cos(60*2019)+sin(60*2019))=\boxed{2017}$ $\newline$ ~ryanbear

Video Solution

https://www.youtube.com/watch?v=aubDsjVFFTc

~bunny1

https://youtu.be/ub6CdxulWfc

~savannahsolver

@@ Line 112: / Line 112: @@
 And <math>f(n)=(An+D)*1^n+B*(\frac{1}{2}-\frac{i\sqrt{3}}{2})^n+C*(\frac{1}{2}+\frac{i\sqrt{3}}{2})^n</math><math>\newline</math>
 Plugging in <math>n=1</math>, <math>n=2</math>, <math>n=3</math>, and <math>n=4</math> results in a system of <math>3</math> linear equations<math>\newline</math>
-Solving them gives <math>A=1, B=\frac{1}{2}-\frac{i\sqrt{3}}{2}, C=\frac{1}{2}+\frac{i\sqrt{3}}{2}, D=1</math><math>\newline</math>
+Solving them gives <math>A=1, B=\frac{1}{2}-\frac{i\sqrt{3}}{2}, C=\frac{1}{2}+\frac{i\sqrt{3}}{2}, D=1</math> (note: you can guess A=1 by answer choices) <math>\newline</math>
 So plugging in <math>n=2018</math> results in <math>2018+1+(\frac{1}{2}-\frac{i\sqrt{3}}{2})^{2019}+(\frac{1}{2}+\frac{i\sqrt{3}}{2})^{2019}=2019+(cos(-60)+sin(-60))^{2019})+(cos(60)+sin(60))^{2019})=2019+(cos(-60*2019)+sin(-60*2019))+(cos(60*2019)+sin(60*2019))=\boxed{2017}</math><math>\newline</math>
 ~ryanbear

2018 AMC 10B (Problems • Answer Key • Resources)
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2018 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
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