Difference between revisions of "2000 AMC 12 Problems/Problem 8"

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<math>\mathrm{(A)}\ 10401 \qquad\mathrm{(B)}\ 19801 \qquad\mathrm{(C)}\ 20201 \qquad\mathrm{(D)}\ 39801 \qquad\mathrm{(E)}\ 40801</math>
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<math>\textbf{(A)}\ 10401 \qquad\textbf{(B)}\ 19801 \qquad\textbf{(C)}\ 20201 \qquad\textbf{(D)}\ 39801 \qquad\textbf{(E)}\ 40801</math>
  
==Solution==
+
==Video:==
 +
https://www.youtube.com/watch?v=HVP6qjKAkjA&t=2s
  
 +
==Solution 1==
  
 +
From the diagrams, we find out that the total number of squares in each figure form an arithmetic sequence: 1,5,13,25. 0 squared + 1 squared = 1 and 1 squared + 2 squared = 5 so the pattern here is (nth term) squared + ((n+1)term) squared= amount of squares in nth figure. When we plug in 100 for n we get 100 squared + 101 squared = 10000 + 10201= 20201 which is answer C.
  
===Solution 2===
+
==Solution 2==
 
 
We can divide up figure <math>n</math> to get the sum of the sum of the first <math>n+1</math> odd numbers and the sum of the first <math>n</math> odd numbers. If you do not see this, here is the example for <math>n=3</math>:
 
 
 
<asy>
 
draw((3,0)--(4,0)--(4,7)--(3,7)--cycle);
 
draw((0,3)--(7,3)--(7,4)--(0,4)--cycle);
 
draw((2,1)--(5,1)--(5,6)--(2,6)--cycle);
 
draw((1,2)--(6,2)--(6,5)--(1,5)--cycle);
 
draw((3,0)--(3,7));
 
</asy>
 
 
 
The sum of the first <math>n</math> odd numbers is <math>n^2</math>, so for figure <math>n</math>, there are <math>(n+1)^2+n^2</math> unit squares. We plug in <math>n=100</math> to get <math>20201</math>, which is choice <math>\boxed{\text{C}}</math>
 
 
 
===Solution 3===
 
  
 
Using the recursion from solution 1, we see that the first differences of <math>4, 8, 12, ...</math> form an arithmetic progression, and consequently that the second differences are constant and all equal to <math>4</math>.  Thus, the original sequence can be generated from a quadratic function.
 
Using the recursion from solution 1, we see that the first differences of <math>4, 8, 12, ...</math> form an arithmetic progression, and consequently that the second differences are constant and all equal to <math>4</math>.  Thus, the original sequence can be generated from a quadratic function.
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Calculating the answer to our problem, <math>f(100) = 20000 + 200 + 1 = 20201</math>, which is choice <math>\boxed{\textbf{(C) }20201}</math>.
 
Calculating the answer to our problem, <math>f(100) = 20000 + 200 + 1 = 20201</math>, which is choice <math>\boxed{\textbf{(C) }20201}</math>.
  
===Solution 4===
+
==Solution 3==
We can see that each figure <math>n</math> has a central box and 4 columns of <math>n</math> boxes on each side of each square. Therefore, at figure 100, there is a central box with 100 boxes on the top, right, left, and bottom. Knowing that each quarter of each figure has a pyramid structure, we know that for each quarter there are <math>\sum_{n=1}^{100} n = 5050</math> squares. <math>4 \cdot 5050 = 20200</math>. Adding in the original center box we have <math> 20200 + 1 = 20201</math> which is answer choice <math> \boxed{\text{C}}</math>
+
We can see that each figure <math>n</math> has a central box and 4 columns of <math>n</math> boxes on each side of each square. Therefore, at figure 100, there is a central box with 100 boxes on the top, right, left, and bottom. Knowing that each quarter of each figure has a pyramid structure, we know that for each quarter there are <math>\sum_{n=1}^{100} n = 5050</math> squares. <math>4 \cdot 5050 = 20200</math>. Adding in the original center box we have <math> 20200 + 1 = \boxed{\textbf{(C) }20201}</math>.
  
==Solution 5==
+
==Solution 4==
 
Let <math>a_n</math> be the number of squares in figure <math>n</math>.  We can easily see that  
 
Let <math>a_n</math> be the number of squares in figure <math>n</math>.  We can easily see that  
 
<cmath>a_0=4\cdot 0+1</cmath>
 
<cmath>a_0=4\cdot 0+1</cmath>
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<cmath>a_2=4\cdot 3+1</cmath>
 
<cmath>a_2=4\cdot 3+1</cmath>
 
<cmath>a_3=4\cdot 6+1.</cmath>
 
<cmath>a_3=4\cdot 6+1.</cmath>
Note that in <math>a_n</math>, the number multiplied by the 4 is the <math>n</math>th triangular number.  Hence, <math>a_{100}=4\cdot \frac{100\cdot 101}{2}+1=20201,</math> which is <math>\boxed{\text{C}}</math>.
+
See that we multiply the number we are on to the next consecutive number.
 +
Note that in <math>a_n</math>, the number multiplied by the 4 is the <math>n</math>th triangular number.  Hence, <math>a_{100}=4\cdot \frac{100\cdot 101}{2}+1=\boxed{\textbf{(C) }20201}</math>.
 +
~ edited by mathlover66
 +
 
 +
==Solution 5==
 +
Let <math>f_n</math> denote the number of unit cubes in a figure. We have
 +
<cmath>f_0=1</cmath>
 +
<cmath>f_1=5</cmath>
 +
<cmath>f_2=13</cmath>
 +
<cmath>f_3=25</cmath>
 +
<cmath>f_4=41</cmath>
 +
<cmath>...</cmath>
 +
 
 +
Computing the difference between the number of cubes in each figure yields
 +
<cmath>4,8,12,16,...</cmath>
 +
It is easy to notice that this is an arithmetic sequence, with the first term being <math>4</math> and the difference being <math>4</math>. Let this sequence be <math>a_n</math>
 +
 
 +
From <math>f_0</math> to <math>f_{100}</math>, the sequence will have <math>100</math> terms. Using the arithmetic sum formula yields
 +
 
 +
<cmath>S_{100}=\frac{100[2\cdot 4+(100-1)4]}{2}</cmath>
 +
<cmath>=50(2\cdot 4+99\cdot 4)</cmath>
 +
<cmath>=50(101\cdot 4)</cmath>
 +
<cmath>=200\cdot 101</cmath>
 +
<cmath>=20200</cmath>
 +
 
 +
So <math>f_{100}=1+20200=\boxed{\textbf{(C) }20201}</math> unit cubes.
 +
 
 +
~ljlbox
 +
 
 +
== Solution 6 (Newton's Forward Differences) ==
 +
We know that 1 and 5 differ by 4, 5 and 13 differ by 8, and 13 and 25 differ by 12. Hence the differences are 4, 8, and 12, resp. And the differences of the differences area all 4. So by Newton's Forward Difference Formula, we get the 100th figure is(because Figure 0 exists) <math>\dbinom{101-1}{0}+4\dbinom{101-1}{1}+4\dbinom{101-1}{2}=20201</math> or <math>\textbf{(C)}</math>
 +
-vsamc
 +
 
 +
 
 +
== Solution 7 (Newton’s little formula) ==
 +
Newton’s little formula states that <math>a_n = A \binom{n-1}{0} + B \binom{n-1}{1} + C \binom{n-1}{2} + \cdots + K \binom{n-1}{m}</math> if first term is <math>a_1</math> and <math>A =</math> first difference, <math>B =</math> second difference, and so on. Hence we apply the formula: <math>a_{100} = 5 \binom{99}{0} + 8 \binom{99}{1} + 4 \binom{99}{2} = \boxed{20201} = \textbf{(C)}.</math>
 +
~Peelybonehead
  
 
==See Also==
 
==See Also==
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{{AMC12 box|year=2000|num-b=7|num-a=9}}
 
{{AMC12 box|year=2000|num-b=7|num-a=9}}
 
{{AMC10 box|year=2000|num-b=11|num-a=13}}
 
{{AMC10 box|year=2000|num-b=11|num-a=13}}
 
+
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 22:31, 13 November 2022

The following problem is from both the 2000 AMC 12 #8 and 2000 AMC 10 #12, so both problems redirect to this page.

Problem

Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?

[asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S); [/asy]


$\textbf{(A)}\ 10401 \qquad\textbf{(B)}\ 19801 \qquad\textbf{(C)}\ 20201 \qquad\textbf{(D)}\ 39801 \qquad\textbf{(E)}\ 40801$

Video:

https://www.youtube.com/watch?v=HVP6qjKAkjA&t=2s

Solution 1

From the diagrams, we find out that the total number of squares in each figure form an arithmetic sequence: 1,5,13,25. 0 squared + 1 squared = 1 and 1 squared + 2 squared = 5 so the pattern here is (nth term) squared + ((n+1)term) squared= amount of squares in nth figure. When we plug in 100 for n we get 100 squared + 101 squared = 10000 + 10201= 20201 which is answer C.

Solution 2

Using the recursion from solution 1, we see that the first differences of $4, 8, 12, ...$ form an arithmetic progression, and consequently that the second differences are constant and all equal to $4$. Thus, the original sequence can be generated from a quadratic function.

If $f(n) = an^2 + bn + c$, and $f(0) = 1$, $f(1) = 5$, and $f(2) = 13$, we get a system of three equations in three variables:

$f(0) = 1$ gives $c = 1$

$f(1) = 5$ gives $a + b + c = 5$

$f(2) = 13$ gives $4a + 2b + c = 13$

Plugging in $c=1$ into the last two equations gives

$a + b = 4$

$4a + 2b = 12$

Dividing the second equation by 2 gives the system:

$a + b = 4$

$2a + b = 6$

Subtracting the first equation from the second gives $a = 2$, and hence $b = 2$. Thus, our quadratic function is:

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

Calculating the answer to our problem, $f(100) = 20000 + 200 + 1 = 20201$, which is choice $\boxed{\textbf{(C) }20201}$.

Solution 3

We can see that each figure $n$ has a central box and 4 columns of $n$ boxes on each side of each square. Therefore, at figure 100, there is a central box with 100 boxes on the top, right, left, and bottom. Knowing that each quarter of each figure has a pyramid structure, we know that for each quarter there are $\sum_{n=1}^{100} n = 5050$ squares. $4 \cdot 5050 = 20200$. Adding in the original center box we have $20200 + 1 = \boxed{\textbf{(C) }20201}$.

Solution 4

Let $a_n$ be the number of squares in figure $n$. We can easily see that \[a_0=4\cdot 0+1\] \[a_1=4\cdot 1+1\] \[a_2=4\cdot 3+1\] \[a_3=4\cdot 6+1.\] See that we multiply the number we are on to the next consecutive number. Note that in $a_n$, the number multiplied by the 4 is the $n$th triangular number. Hence, $a_{100}=4\cdot \frac{100\cdot 101}{2}+1=\boxed{\textbf{(C) }20201}$. ~ edited by mathlover66

Solution 5

Let $f_n$ denote the number of unit cubes in a figure. We have \[f_0=1\] \[f_1=5\] \[f_2=13\] \[f_3=25\] \[f_4=41\] \[...\]

Computing the difference between the number of cubes in each figure yields \[4,8,12,16,...\] It is easy to notice that this is an arithmetic sequence, with the first term being $4$ and the difference being $4$. Let this sequence be $a_n$

From $f_0$ to $f_{100}$, the sequence will have $100$ terms. Using the arithmetic sum formula yields

\[S_{100}=\frac{100[2\cdot 4+(100-1)4]}{2}\] \[=50(2\cdot 4+99\cdot 4)\] \[=50(101\cdot 4)\] \[=200\cdot 101\] \[=20200\]

So $f_{100}=1+20200=\boxed{\textbf{(C) }20201}$ unit cubes.

~ljlbox

Solution 6 (Newton's Forward Differences)

We know that 1 and 5 differ by 4, 5 and 13 differ by 8, and 13 and 25 differ by 12. Hence the differences are 4, 8, and 12, resp. And the differences of the differences area all 4. So by Newton's Forward Difference Formula, we get the 100th figure is(because Figure 0 exists) $\dbinom{101-1}{0}+4\dbinom{101-1}{1}+4\dbinom{101-1}{2}=20201$ or $\textbf{(C)}$ -vsamc


Solution 7 (Newton’s little formula)

Newton’s little formula states that $a_n = A \binom{n-1}{0} + B \binom{n-1}{1} + C \binom{n-1}{2} + \cdots + K \binom{n-1}{m}$ if first term is $a_1$ and $A =$ first difference, $B =$ second difference, and so on. Hence we apply the formula: $a_{100} = 5 \binom{99}{0} + 8 \binom{99}{1} + 4 \binom{99}{2} = \boxed{20201} = \textbf{(C)}.$ ~Peelybonehead

See Also

2000 AMC 12 (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
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
2000 AMC 10 (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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

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