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

(Reformatted answer choices and made sols for professional. In addition, I prioritize the nonbashing solutions, and decide to rewrite Sol 2 somehow. Credits will retain to original authors. Let me know if you are unhappy with it.)
(→‎Solution 4)
(15 intermediate revisions by 3 users not shown)
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==Solution 1 (Algebra)==
 
==Solution 1 (Algebra)==
−
<b>IN CONSTRUCTION</b>
+
For all integers <math>n \geq 7,</math> note that
−
 
+
<cmath>\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*}</cmath>
 +
It follows that
 +
<cmath>\begin{align*}
 +
f(2018)&=f(2012)+6 \\
 +
&=f(2006)+12 \\
 +
&=f(2000)+18 \\
 +
& \ \vdots \\
 +
&=f(2)+2016 \\
 +
&=\boxed{\textbf{(B) } 2017}.
 +
\end{align*}</cmath>
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
 
==Solution 2 (Algebra)==
 
==Solution 2 (Algebra)==
−
<b>IN CONSTRUCTION</b>
+
For all integers <math>n\geq3,</math> we rearrange the given equation: <cmath>f(n)-f(n-1)+f(n-2)=n. \hspace{28.25mm}(1)</cmath>
 +
For all integers <math>n\geq4,</math> it follows that <cmath>f(n-1)-f(n-2)+f(n-3)=n-1. \hspace{15mm}(2)</cmath>
 +
For all integers <math>n\geq4,</math> we add <math>(1)</math> and <math>(2):</math> <cmath>f(n)+f(n-3)=2n-1. \hspace{38.625mm}(3)</cmath>
 +
For all integers <math>n\geq7,</math> it follows that <cmath>f(n-3)+f(n-6)=2n-7. \hspace{32mm}(4)</cmath>
 +
For all integers <math>n\geq7,</math> we subtract <math>(4)</math> from <math>(3):</math> <cmath>f(n)-f(n-6)=6. \hspace{47.5mm}(5)</cmath>
 +
From <math>(5),</math> we have the following system of <math>336</math> equations:
 +
<cmath>\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*}</cmath>
 +
We add these equations up to get <cmath>f(2018)-f(2)=6\cdot336=2016,</cmath> from which <math>f(2018)=f(2)+2016=\boxed{\textbf{(B) } 2017}.</math>
  
 
~AopsUser101 ~MRENTHUSIASM
 
~AopsUser101 ~MRENTHUSIASM
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Preamble: In this solution, we define the sequence <math>A</math> to satisfy <math>a_n = f(n),</math> where <math>a_n</math> represents the <math>n</math>th term of the sequence <math>A.</math> 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.  
 
Preamble: In this solution, we define the sequence <math>A</math> to satisfy <math>a_n = f(n),</math> where <math>a_n</math> represents the <math>n</math>th term of the sequence <math>A.</math> 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 <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 <cmath>\begin{cases} a_{n+1} = a_{n} - a_{n-1} + (n+1) \\ a_{n} = a_{n-1} - a_{n-2} + n. \end{cases}</cmath> Notice that subtracting the second equation from the first, we see that <math>b_{n} = b_{n-1} - b_{n-2} + 1.</math>  
+
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.  
 
Now, there are two ways to finish.  
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Finish Method #2: Notice that <math>a_{2018} = a_{2017} - a_{2016} + 2018 = B_{2016} + 2018 = -1 + 2018 = \boxed{\textbf{(B) } 2017}.</math>  
 
Finish Method #2: Notice that <math>a_{2018} = a_{2017} - a_{2016} + 2018 = B_{2016} + 2018 = -1 + 2018 = \boxed{\textbf{(B) } 2017}.</math>  
  
−
~ Professor-Mom
+
~Professor-Mom
  
−
==Solution 4 (Bash)==
+
==Solution 4 (Pattern)==
 
Start out by listing some terms of the sequence.  
 
Start out by listing some terms of the sequence.  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
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f(14)&=13 \\
 
f(14)&=13 \\
 
f(15)&=15 \\
 
f(15)&=15 \\
−
&\cdots
+
& \ \vdots
 
\end{align*}</cmath>
 
\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>.
 
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>.
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f(2018)&=\boxed{\textbf{(B) } 2017}.
 
f(2018)&=\boxed{\textbf{(B) } 2017}.
 
\end{align*}</cmath>
 
\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>
  
 
==Video Solution==
 
==Video Solution==

Revision as of 20:27, 14 November 2022

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 $f(2017)=f(2018)=\boxed{\textbf{(B) } 2017}.$

Video Solution

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

~bunny1

https://youtu.be/ub6CdxulWfc

~savannahsolver

See Also

2018 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2018 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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