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

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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{(D) 78}</math>
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==Solution 4==
  
 
== Additional Comment==
 
== Additional Comment==

Revision as of 16:00, 27 December 2019

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

~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 $a_{12}$. $a_{12} = a_6+a_6+6\cdot 6 = 21+21+36= \boxed{(D) 78}$

Solution 4

Additional Comment

This is also the formula for the triangular numbers $T_n=1+2+3+...+n$, as seen in Solution 2

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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