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Difference between revisions of "2002 AMC 10B Problems/Problem 23"

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== Solution 1 ==
 
== Solution 1 ==
  
−
First of all, write <math>a_3</math> and <math>a_4</math> in terms of <math>a_2.</math>
+
When <math>m=1</math>, <math>a_{n+1}=1+a_n+n</math>. Hence,
−
<cmath>\begin{align*} a_3 &= a_{1+2} = 1 + a_2 + 2 = a_2 + 3\\
+
<cmath>a_{2}=1+a_1+1</cmath>
−
a_4 &= a_{2+2} = a_2 + a_2 + 4 = 2a_2 + 4 \end{align*}</cmath>
+
<cmath>a_{3}=1+a_2+2</cmath>
 +
<cmath>a_{4}=1+a_3+3</cmath>
 +
<cmath>\dots</cmath>
 +
<cmath>a_{12}=1+a_{11}+11</cmath>
 +
Adding these equations up, we have that <math>a_{12}=12+(1+2+3+...+11)=\boxed{\mathrm{(D) \ } 78}</math>
  
−
<math>a_6</math> can be represented by <math>a_2</math> in <math>2</math> different ways.
+
~AopsUser101
  
−
<cmath>\begin{align*} a_{2+4} &= a_2 + a_4 + 8 = a_2 + 2a_2 + 4 + 8 = 3a_2 + 12\\
+
== Solution 2 ==
−
a_{3+3} &= 2a_3 + 9 = 6 + 2a_2 + 9 = 2a_2 + 15\\
+
 
−
\end{align*}</cmath>
+
Substituting <math>n=1</math> into <math>a_{m+n}=a_m+a_n+mn</math>: <math>a_{m+1}=a_m+a_{1}+m</math>.
 +
 
 +
Since <math>a_1 = 1</math>, <math>a_{m+1}=a_m+m+1</math>.
 +
 
 +
Therefore, <math>a_m = a_{m-1} + m, a_{m-1}=a_{m-2}+(m-1), a_{m-2} = a_{m-3} + (m-2)</math>, and so on until <math>a_2 = a_1 + 2</math>.
  
−
Since both are equal to <math>a_6,</math> you can set them equal to each other.
+
Adding the Left Hand Sides of all of these equations gives <math>a_m + a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_2</math>.
  
−
<cmath>\begin{align*} 3a_2 + 12 &= 2a_2 + 15\\
+
Adding the Right Hand Sides of these equations gives
−
a_2 &= 3 \end{align*}</cmath>
 
  
−
Substitute the value of <math>a_2</math> back into <math>a_6,</math> and substitute that into <math>a_{12}.</math>
+
<math>(a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_1) + (m + (m-1) + (m-2) + \cdots + 2)</math>.
  
−
<cmath>a_6 = 2a_2+15 = 6+15 = 21</cmath>
+
These two expressions must be equal; hence <math>a_m + a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_2 = (a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_1) + (m + (m-1) + (m-2) + \cdots + 2)</math> and <math>a_m = a_1 + (m + (m-1) + (m-2) + \cdots + 2)</math>.
−
<cmath>a_{12} = a_{6+6} = a_6+a_6+36 = 21+21+36 = \boxed{\mathrm{(D) \ } 78}</cmath>
 
  
−
== Solution 2 ==
+
Substituting <math>a_1 = 1</math>: <math>a_m = 1 + (m + (m-1) + (m-2) + \cdots + 2) = 1+2+3+4+ \cdots +m = \frac{(m+1)(m)}{2}</math>.
  
−
Substituting <math>n=1</math> into <math>a_{m+n}=a_m+a_n+mn</math>: <math>a_{m+1}=a_m+a_{1}+m</math>. Since <math>a_1 = 1</math>, <math>a_{m+1}=a_m+m+1</math>. Therefore, <math>a_m = a_{m-1} + m, a_{m-1}=a_{m-2}+(m-1), a_{m-2} = a_{m-3} + (m-2)</math>, and so on until <math>a_2 = a_1 + 2</math>. Adding the Left Hand Sides of all of these equations gives <math>a_m + a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_2</math>; adding the Right Hand Sides of these equations gives <math>(a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_1) + (m + (m-1) + (m-2) + \cdots + 2)</math>. These two expressions must be equal; hence <math>a_m + a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_2 = (a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_1) + (m + (m-1) + (m-2) + \cdots + 2)</math> and <math>a_m = a_1 + (m + (m-1) + (m-2) + \cdots + 2)</math>. Substituting <math>a_1 = 1</math>: <math>a_m = 1 + (m + (m-1) + (m-2) + \cdots + 2) = 1+2+3+4+ \cdots +m = \frac{(m+1)(m)}{2}</math>. Thus we have a general formula for <math>a_m</math> and substituting <math>m=12</math>: <math>a_{12} = \frac{(13)(12)}{2} = (13)(6) = \boxed{\mathrm{(D) \ } 78}</math>.
+
Thus we have a general formula for <math>a_m</math> and substituting <math>m=12</math>: <math>a_{12} = \frac{(13)(12)}{2} = (13)(6) = \boxed{\mathrm{(D)} 78}</math>.
  
 
==Solution 3==
 
==Solution 3==
  
−
We can literally just plug stuff in. No prerequisite is actually said in the sequence. Since <math>a_{m+n} = a_m+a_n +mn</math>, we know <math>a_2=a_1+a_1+1\cdot1=1+1+1=3</math>. After this, we can use <math>a_2</math> to find <math>a_4</math>. <math>a_4=a_2+a_2+2\cdot 2 = 3+3+4 = 10</math>. Now, we can use <math>a_2</math> and <math>a_4</math> to find <math>a_6</math>, or <math>a_6=a_4+a_2+4\cdot 2 = 10+3+8=21</math>. Lastly, we can use <math>a_6</math> to find <math>a_{12}</math>. <math>a_{12} = a_6+a_6+6\cdot 6 = 21+21+36= \boxed{(D) 78}</math>
+
We can literally just plug stuff in. No prerequisite is actually said in the sequence. Since <math>a_{m+n} = a_m+a_n +mn</math>, we know <math>a_2=a_1+a_1+1\cdot1=1+1+1=3</math>. After this, we can use <math>a_2</math> to find <math>a_4</math>. <math>a_4=a_2+a_2+2\cdot 2 = 3+3+4 = 10</math>. Now, we can use <math>a_2</math> and <math>a_4</math> to find <math>a_6</math>, or <math>a_6=a_4+a_2+4\cdot 2 = 10+3+8=21</math>. Lastly, we can use <math>a_6</math> to find <math>a_{12}</math>. <math>a_{12} = a_6+a_6+6\cdot 6 = 21+21+36= \boxed{\mathrm{(D) \ } 78}</math>
 +
 
 +
==Solution 4==
 +
 
 +
We can set <math>n</math> equal to <math>m</math>, so we can say that
 +
<cmath>a_{m + m} = a_m + a_m + m*m</cmath>
 +
<cmath>a_{2m} = 2a_m + m^2</cmath>
 +
 
 +
We set <math>2m = 12</math>, we get <math>m = 6</math>.
 +
<cmath>a_{12} = 2a_6 + 36</cmath>
 +
 
 +
We set <math>2m = 6</math>m, we get <math>m = 3</math>.
 +
<cmath>a_6 = 2a_3 + 9</cmath>
 +
 
 +
Solving for <math>a_3</math> is easy, just direct substitution.
 +
<cmath>a_2 = 1 + 1 + 1 = 3</cmath>
 +
<cmath>a_3 = a_{2 + 1} = 3 + 1 + 2 = 6</cmath>
 +
 
 +
Substituting, we get
 +
<cmath>a_6 = 2(6) + 9 = 21</cmath>
 +
<cmath>a_{12} = 2(21) + 36 = 78</cmath>
 +
 
 +
Thus, the answer is <math>\boxed{D}</math>.
 +
 
 +
~ euler123
 +
 
 +
== Solution 5 ==
 +
 
 +
Note that the sequence of triangular numbers <math>T_n=1+2+3+...+n</math> satisfies these conditions. It is immediately obvious that it satisfies <math>a_1=1</math>, and <math>a_{m+n}=a_m+a_n+mn</math> can be visually proven with the diagram below.
 +
 
 +
<asy>
 +
for(int i=5; i > 0; --i) {
 +
for(int j=0; j < i; ++j) {
 +
draw(circle((j+(5-i)/2,(5-i)*sqrt(3)/2),.2));
 +
};
 +
};
 +
path m1 = brace((2,-.3),(0,-.3),.2);
 +
draw(m1);
 +
label("$m$",m1,S);
 +
 
 +
path n1 = brace((4,-.3),(3,-.3),.2);
 +
draw(n1);
 +
label("$n$",n1,S);
 +
 
 +
 
 +
draw((-.2*sqrt(3),-.2)--(2+.2*sqrt(3),-.2)--(1,.4+sqrt(3))--cycle);
 +
label("$T_m$",(1,1/3*sqrt(3)));
 +
 
 +
draw((3-.2*sqrt(3),-.2)--(4+.2*sqrt(3),-.2)--(3.5,.4+.5*sqrt(3))--cycle);
 +
label("$T_n$",(3.5,.5/3*sqrt(3)));
 +
 
 +
 
 +
path m2 = brace((2+.15*sqrt(3),.15+2*sqrt(3)),(3+.15*sqrt(3),.15+sqrt(3)),.2);
 +
draw(m2);
 +
label("$m$",m2,(.5*sqrt(3),.5));
 +
 
 +
path n2 = brace((1.5-.15*sqrt(3),.15+1.5*sqrt(3)),(2-.15*sqrt(3),.15+2*sqrt(3)),.2);
 +
draw(n2);
 +
label("$n$",n2,(-.5*sqrt(3),.5));
 +
 
 +
 
 +
draw((2.5,-.4+.5*sqrt(3))--(3+.4/3*sqrt(3),sqrt(3))--(2,.4+2*sqrt(3))--(1.5-.4/3*sqrt(3),1.5*sqrt(3))--cycle);
 +
label("$mn$",(2.25,1.25*sqrt(3)));
 +
</asy>
 +
 
 +
This means that we can use the triangular number formula <math>T_n = \frac{n(n+1)}{2}</math>, so the answer is <math>T_{12} = \frac{12(12+1)}{2} = \boxed{\mathrm{(D) \ } 78}</math>.
 +
 
 +
 
 +
~[[User:emerald_block|emerald_block]]
 +
 
  
−
== Additional Comment==
+
== Solution 6 ==
 +
Follow solution 4 with plugging in m=n, but for the last part since since the subscript is 2n, and we aren't given anything that has odd numbers ( without substitution into the original equation), we need to find two numbers that add up to 12, which result from the multiplication of two even numbers. We realize that 4+8 works, since 4=2x2 and 8=4x2. We then proceed in substituting n=2 and n=4 into the equation, then add to obtain 78.
 +
~Charmainema07292010
  
−
This is also the formula for the triangular numbers <math>T_n=1+2+3+...+n</math>, as seen in Solution 2
+
== Video Solution ==
 +
https://www.youtube.com/watch?v=zraGzYAh0uM  ~David
  
 
== See also ==
 
== See also ==
 
{{AMC10 box|year=2002|ab=B|num-b=22|num-a=24}}
 
{{AMC10 box|year=2002|ab=B|num-b=22|num-a=24}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 05:38, 23 October 2023

Problem 23

Let $\{a_k\}$ be a sequence of integers such that $a_1=1$ and $a_{m+n}=a_m+a_n+mn,$ for all positive integers $m$ and $n.$ Then $a_{12}$ is

$\mathrm{(A) \ } 45\qquad \mathrm{(B) \ } 56\qquad \mathrm{(C) \ } 67\qquad \mathrm{(D) \ } 78\qquad \mathrm{(E) \ } 89$

Solution 1

When $m=1$, $a_{n+1}=1+a_n+n$. Hence, \[a_{2}=1+a_1+1\] \[a_{3}=1+a_2+2\] \[a_{4}=1+a_3+3\] \[\dots\] \[a_{12}=1+a_{11}+11\] Adding these equations up, we have that $a_{12}=12+(1+2+3+...+11)=\boxed{\mathrm{(D) \ } 78}$

~AopsUser101

Solution 2

Substituting $n=1$ into $a_{m+n}=a_m+a_n+mn$: $a_{m+1}=a_m+a_{1}+m$.

Since $a_1 = 1$, $a_{m+1}=a_m+m+1$.

Therefore, $a_m = a_{m-1} + m, a_{m-1}=a_{m-2}+(m-1), a_{m-2} = a_{m-3} + (m-2)$, and so on until $a_2 = a_1 + 2$.

Adding the Left Hand Sides of all of these equations gives $a_m + a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_2$.

Adding the Right Hand Sides of these equations gives

$(a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_1) + (m + (m-1) + (m-2) + \cdots + 2)$.

These two expressions must be equal; hence $a_m + a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_2 = (a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_1) + (m + (m-1) + (m-2) + \cdots + 2)$ and $a_m = a_1 + (m + (m-1) + (m-2) + \cdots + 2)$.

Substituting $a_1 = 1$: $a_m = 1 + (m + (m-1) + (m-2) + \cdots + 2) = 1+2+3+4+ \cdots +m = \frac{(m+1)(m)}{2}$.

Thus we have a general formula for $a_m$ and substituting $m=12$: $a_{12} = \frac{(13)(12)}{2} = (13)(6) = \boxed{\mathrm{(D)} 78}$.

Solution 3

We can literally just plug stuff in. No prerequisite is actually said in the sequence. Since $a_{m+n} = a_m+a_n +mn$, we know $a_2=a_1+a_1+1\cdot1=1+1+1=3$. After this, we can use $a_2$ to find $a_4$. $a_4=a_2+a_2+2\cdot 2 = 3+3+4 = 10$. Now, we can use $a_2$ and $a_4$ to find $a_6$, or $a_6=a_4+a_2+4\cdot 2 = 10+3+8=21$. Lastly, we can use $a_6$ to find $a_{12}$. $a_{12} = a_6+a_6+6\cdot 6 = 21+21+36= \boxed{\mathrm{(D) \ } 78}$

Solution 4

We can set $n$ equal to $m$, so we can say that \[a_{m + m} = a_m + a_m + m*m\] \[a_{2m} = 2a_m + m^2\]

We set $2m = 12$, we get $m = 6$. \[a_{12} = 2a_6 + 36\]

We set $2m = 6$m, we get $m = 3$. \[a_6 = 2a_3 + 9\]

Solving for $a_3$ is easy, just direct substitution. \[a_2 = 1 + 1 + 1 = 3\] \[a_3 = a_{2 + 1} = 3 + 1 + 2 = 6\]

Substituting, we get \[a_6 = 2(6) + 9 = 21\] \[a_{12} = 2(21) + 36 = 78\]

Thus, the answer is $\boxed{D}$.

~ euler123

Solution 5

Note that the sequence of triangular numbers $T_n=1+2+3+...+n$ satisfies these conditions. It is immediately obvious that it satisfies $a_1=1$, and $a_{m+n}=a_m+a_n+mn$ can be visually proven with the diagram below.

[asy] for(int i=5; i > 0; --i) { for(int j=0; j < i; ++j) { draw(circle((j+(5-i)/2,(5-i)*sqrt(3)/2),.2)); }; }; path m1 = brace((2,-.3),(0,-.3),.2); draw(m1); label("$m$",m1,S); path n1 = brace((4,-.3),(3,-.3),.2); draw(n1); label("$n$",n1,S); draw((-.2*sqrt(3),-.2)--(2+.2*sqrt(3),-.2)--(1,.4+sqrt(3))--cycle); label("$T_m$",(1,1/3*sqrt(3))); draw((3-.2*sqrt(3),-.2)--(4+.2*sqrt(3),-.2)--(3.5,.4+.5*sqrt(3))--cycle); label("$T_n$",(3.5,.5/3*sqrt(3))); path m2 = brace((2+.15*sqrt(3),.15+2*sqrt(3)),(3+.15*sqrt(3),.15+sqrt(3)),.2); draw(m2); label("$m$",m2,(.5*sqrt(3),.5)); path n2 = brace((1.5-.15*sqrt(3),.15+1.5*sqrt(3)),(2-.15*sqrt(3),.15+2*sqrt(3)),.2); draw(n2); label("$n$",n2,(-.5*sqrt(3),.5)); draw((2.5,-.4+.5*sqrt(3))--(3+.4/3*sqrt(3),sqrt(3))--(2,.4+2*sqrt(3))--(1.5-.4/3*sqrt(3),1.5*sqrt(3))--cycle); label("$mn$",(2.25,1.25*sqrt(3))); [/asy]

This means that we can use the triangular number formula $T_n = \frac{n(n+1)}{2}$, so the answer is $T_{12} = \frac{12(12+1)}{2} = \boxed{\mathrm{(D) \ } 78}$.


~emerald_block


Solution 6

Follow solution 4 with plugging in m=n, but for the last part since since the subscript is 2n, and we aren't given anything that has odd numbers ( without substitution into the original equation), we need to find two numbers that add up to 12, which result from the multiplication of two even numbers. We realize that 4+8 works, since 4=2x2 and 8=4x2. We then proceed in substituting n=2 and n=4 into the equation, then add to obtain 78. ~Charmainema07292010

Video Solution

https://www.youtube.com/watch?v=zraGzYAh0uM ~David

See also

2002 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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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