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

Revision as of 01:11, 11 September 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

$\begin{align*} f(n)&=f(n-1)-f(n-2)+n \\ f(n-1)&=f(n-2)-f(n-3)+n-1 \end{align*}$ Subtracting those two and rearranging gives $\begin{align*} f(n)-2f(n-1)+2f(n-2)-f(n-3)&=1 \\ f(n-1)-2f(n-2)+2f(n-3)-f(n-4)&=1 \end{align*}$ Subtracting those two gives $f(n)-3f(n-1)+4f(n-2)-3f(n-3)+f(n-4)=0.$

The characteristic polynomial is $x^4-3x^3+4x^2-3x+1=0.$

$x=1$ is a root, so using synthetic division results in $(x-1)(x^3-2x^2+2x-1)=0.$

$x=1$ is a root, so using synthetic division results in $(x-1)^2(x^2-x+1)=0.$

$x^2-x+1=0$ has roots $x=\frac{1}{2}\pm\frac{i\sqrt{3}}{2}.$

And $f(n)=(An+D)\cdot1^n+B\cdot\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)^n+C\cdot\left(\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)^n.$ Plugging in $n=1$ , $n=2$ , $n=3$ , and $n=4$ results in a system of $4$ 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 that you can guess $A=1$ by answer choices.

So plugging in $n=2018$ results in $\begin{align*} 2018+1+\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)^{2019}+\left(\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)^{2019}&=2019+(\cos(-60^{\circ})+\sin(-60^{\circ}))^{2019})+(\cos(60^{\circ})+\sin(60^{\circ}))^{2019}) \\ &=2019+(\cos(-60^{\circ}\cdot2019)+\sin(-60^{\circ}\cdot2019))+(\cos(60^{\circ}\cdot2019)+sin(60^{\circ}\cdot2019)) \\ &=\boxed{\textbf{(B) } 2017}. \end{align*}$ ~ryanbear

Solution 7

We utilize patterns to solve this equation: $\begin{align*} f(3)&=3, \\ f(4)&=6, \\ f(5)&=8, \\ f(6)&=8, \\ f(7)&=7, \\ f(8)&=8. \end{align*}$ We realize that the pattern repeats itself. For every six terms, there will be four terms that we repeat, and two terms that we don't repeat. We will exclude the first two for now, because they don't follow this pattern.

First, we need to know whether or not $2016$ is part of the skip or repeat. We notice that $f(6),f(12), \ldots,f(6n)$ all satisfy $6+6(n-1)=n,$ and we know that $2016$ satisfies this, leaving $n=336.$ Therefore, we know that $2016$ is part of the repeat section. But what number does it repeat?

We know that the repeat period is $2,$ and it follows that pattern of $1,1,8,8,7,7.$ Again, since $f(6) = f(5)$ and so on for the repeat section, $f(2016)=f(2015),$ so we don't need to worry about which one, since it repeats with period $2.$ We see that the repeat pattern of $f(6),f(12),\ldots,f(6n)$ follows $8,14,20,$ it is an arithmetic sequence with common difference $6.$ Therefore, $2016$ is the $335$ th term of this, but including $1,$ it is $336\cdot6+1=\boxed{\textbf{(B) } 2017}.$

~CharmaineMa07292010

Video Solution

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

~bunny1

https://youtu.be/ub6CdxulWfc

~savannahsolver

@@ Line 23: / Line 23: @@
 &=f(2006)+12 \\
 &=f(2000)+18 \\
-&=\cdots \\
+& \ \vdots \\
 &=f(2)+2016 \\
 &=\boxed{\textbf{(B) } 2017}.
@@ Line 40: / Line 40: @@
 f(2012)-f(2006)&=6, \\
 f(2006)-f(2000)&=6, \\
-&\cdots \\
+& \ \vdots \\
 f(8)-f(2)&=6.
 \end{align*}</cmath>
@@ Line 53: / Line 53: @@
 To begin, we consider the sequence <math>B</math> formed when we take the difference of consecutive terms between <math>A.</math> Define <math>b_n = a_{n+1} - a_n.</math> Notice that for <math>n \ge 4,</math> we have <center><math>\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}</math></center> Notice that subtracting the second equation from the first, we see that <math>b_{n} = b_{n-1} - b_{n-2} + 1.</math>
-If you didn’t notice that <math>B</math> repeated directly in the solution above, you could also, possibly more naturally, take the finite differences of the sequence <math>b_n</math> so that you could define <math>c_n = b_{n+1} - b_n.</math> Using a similar method as above through reindexing and then subtracting, you could find that <math>c_n = c_{n-1} - c_{n-2}.</math> The sum of any six consecutive terms of a sequence which satisfies such a recursion is <math>0,</math> in which you have that <math>b_{n} = b_{n+6}.</math> In the case in which finite differences didn’t reduce to such a special recursion, you could still find the first few terms of <math>C</math> 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 <math>C</math> looks like <cmath>\underbrace{2, 1, -1, -2, -1, 1,}_{\text{cycle period}} 2, 1, -1, -2, -1, 1, \cdots</cmath> in which the same result follows.
+If you didn’t notice that <math>B</math> repeated directly in the solution above, you could also, possibly more naturally, take the finite differences of the sequence <math>b_n</math> so that you could define <math>c_n = b_{n+1} - b_n.</math> Using a similar method as above through reindexing and then subtracting, you could find that <math>c_n = c_{n-1} - c_{n-2}.</math> The sum of any six consecutive terms of a sequence which satisfies such a recursion is <math>0,</math> in which you have that <math>b_{n} = b_{n+6}.</math> In the case in which finite differences didn’t reduce to such a special recursion, you could still find the first few terms of <math>C</math> 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 <math>C</math> looks like <cmath>\underbrace{2, 1, -1, -2, -1, 1,}_{\text{cycle period}} 2, 1, -1, -2, -1, 1, \ldots</cmath> in which the same result follows.
-Using the fact that <math>B</math> repeats every six terms, this motivates us to look at the sequence <math>B</math> more carefully. Doing so, we see that <math>B</math> looks like <cmath>\underbrace{2, 3, 2, 0, -1, 0,}_{\text{cycle period}} 2, 3, 2, 0, -1, 0, \cdots</cmath> (If you tried pattern finding on sequence <math>B</math> directly, you could also arrive at this result, although I figured defining a second sequence based on finite differences was more motivated.)
+Using the fact that <math>B</math> repeats every six terms, this motivates us to look at the sequence <math>B</math> more carefully. Doing so, we see that <math>B</math> looks like <cmath>\underbrace{2, 3, 2, 0, -1, 0,}_{\text{cycle period}} 2, 3, 2, 0, -1, 0, \ldots</cmath> (If you tried pattern finding on sequence <math>B</math> 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.
@@ Line 65: / Line 65: @@
 ~Professor-Mom
-==Solution 4 (Bash)==
+==Solution 4 (Pattern)==
 Start out by listing some terms of the sequence.
 <cmath>\begin{align*}
@@ Line 83: / Line 83: @@
 f(14)&=13 \\
 f(15)&=15 \\
-&\cdots
+& \ \vdots
 \end{align*}</cmath>
 Notice that <math>f(n)=n</math> whenever <math>n</math> is an odd multiple of <math>3</math>, and the pattern of numbers that follow will always be <math>+2</math>, <math>+3</math>, <math>+2</math>, <math>+0</math>, <math>-1</math>, <math>+0</math>.
@@ Line 95: / Line 95: @@
 f(2018)&=\boxed{\textbf{(B) } 2017}.
 \end{align*}</cmath>
-minor edits by bunny1
-==Solution 5 (Bash)==
+==Solution 5 (Pattern)==
-Writing out the first few values, we get:
+Writing out the first few values, we get
-<math>1,1,3,6,8,8,7,7,9,12,14,14,13,13,15,18,20,20,19,19,\ldots</math>. Examining, we see that every number <math>x</math> where <math>x \equiv 1\pmod 6</math> has <math>f(x)=x</math>, <math>f(x+1)=f(x)=x</math>, and <math>f(x-1)=f(x-2)=x+1</math>. The greatest number that's <math>1\pmod{6}</math> and less <math>2018</math> is <math>2017</math>, so we have <math>f(2017)=f(2018)=\boxed{\textbf{(B) } 2017}</math>.
+<cmath>1,1,3,6,8,8,7,7,9,12,14,14,13,13,15,18,20,20,19,19,\ldots.</cmath> We see that every number <math>x</math> where <math>x \equiv 1\pmod 6</math> has <math>f(x)=x,f(x+1)=f(x)=x,</math> and <math>f(x-1)=f(x-2)=x+1.</math> The greatest number that's <math>1\pmod{6}</math> and less than <math>2018</math> is <math>2017,</math> so we have <math>f(2017)=f(2018)=\boxed{\textbf{(B) } 2017}.</math>
+==Solution 6==
+<cmath>\begin{align*}
+f(n)&=f(n-1)-f(n-2)+n \\
+f(n-1)&=f(n-2)-f(n-3)+n-1
+\end{align*}</cmath>
+Subtracting those two and rearranging gives
+<cmath>\begin{align*}
+f(n)-2f(n-1)+2f(n-2)-f(n-3)&=1 \\
+f(n-1)-2f(n-2)+2f(n-3)-f(n-4)&=1
+\end{align*}</cmath>
+Subtracting those two gives <math>f(n)-3f(n-1)+4f(n-2)-3f(n-3)+f(n-4)=0.</math>
+The characteristic polynomial is <math>x^4-3x^3+4x^2-3x+1=0.</math>
+<math>x=1</math> is a root, so using synthetic division results in <math>(x-1)(x^3-2x^2+2x-1)=0.</math>
+<math>x=1</math> is a root, so using synthetic division results in <math>(x-1)^2(x^2-x+1)=0.</math>
+<math>x^2-x+1=0</math> has roots <math>x=\frac{1}{2}\pm\frac{i\sqrt{3}}{2}.</math>
+And <cmath>f(n)=(An+D)\cdot1^n+B\cdot\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)^n+C\cdot\left(\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)^n.</cmath>
+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>4</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> Note that you can guess <math>A=1</math> by answer choices.
+So plugging in <math>n=2018</math> results in
+<cmath>\begin{align*}
++1+\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)^{2019}+\left(\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)^{2019}&=2019+(\cos(-60^{\circ})+\sin(-60^{\circ}))^{2019})+(\cos(60^{\circ})+\sin(60^{\circ}))^{2019}) \\
+&=2019+(\cos(-60^{\circ}\cdot2019)+\sin(-60^{\circ}\cdot2019))+(\cos(60^{\circ}\cdot2019)+sin(60^{\circ}\cdot2019)) \\
+&=\boxed{\textbf{(B) } 2017}.
+\end{align*}</cmath>
+~ryanbear
+==Solution 7==
+We utilize patterns to solve this equation:
+<cmath>\begin{align*}
+f(3)&=3, \\
+f(4)&=6, \\
+f(5)&=8, \\
+f(6)&=8, \\
+f(7)&=7, \\
+f(8)&=8.
+\end{align*}</cmath>
+We realize that the pattern repeats itself. For every six terms, there will be four terms that we repeat, and two terms that we don't repeat. We will exclude the first two for now, because they don't follow this pattern.
+First, we need to know whether or not <math>2016</math> is part of the skip or repeat. We notice that <math>f(6),f(12), \ldots,f(6n)</math> all satisfy <math>6+6(n-1)=n,</math> and we know that <math>2016</math> satisfies this, leaving <math>n=336.</math> Therefore, we know that <math>2016</math> is part of the repeat section. But what number does it repeat?
+We know that the repeat period is <math>2,</math> and it follows that pattern of <math>1,1,8,8,7,7.</math> Again, since <math>f(6) = f(5)</math> and so on  for the repeat section, <math>f(2016)=f(2015),</math> so we don't need to worry about which one, since it repeats with period <math>2.</math> We see that the repeat pattern of <math>f(6),f(12),\ldots,f(6n)</math> follows <math>8,14,20,</math> it is an arithmetic sequence with common difference <math>6.</math> Therefore, <math>2016</math> is the <math>335</math>th term of this, but including <math>1,</math> it is <math>336\cdot6+1=\boxed{\textbf{(B) } 2017}.</math>
+~CharmaineMa07292010
 ==Video Solution==

2018 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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2018 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
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