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Difference between revisions of "2008 AMC 10B Problems/Problem 13"

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==Problem==
 
==Problem==
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{{problem}}
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For each positive integer <math>n</math>, the mean of the first <math>n</math> terms of a sequence is <math>n</math>. What is the <math>2008^{\text{th}}</math> term of the sequence?
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<math>\mathrm{(A)}\ {{{2008}}} \qquad \mathrm{(B)}\ {{{4015}}} \qquad \mathrm{(C)}\ {{{4016}}} \qquad \mathrm{(D)}\ {{{4,030,056}}} \qquad \mathrm{(E)}\ {{{4,032,064}}}</math>
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==Solution 1==
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Since the mean of the first <math>n</math> terms is <math>n</math>, the sum of the first <math>n</math> terms is <math>n^2</math>. Thus, the sum of the first <math>2007</math> terms is <math>2007^2</math> and the sum of the first <math>2008</math> terms is <math>2008^2</math>. Hence, the <math>2008^{\text{th}}</math> term of the sequence is <math>2008^2-2007^2=(2008+2007)(2008-2007)=4015\Rightarrow \boxed{\textbf{(B) 4015}}</math>
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Note that <math>n^2</math> is the sum of the first n odd integers.
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==Solution 2==
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Let <math>a_1, a_2, a_3, \cdots, a_n</math> be the terms of the sequence. We know <math>\frac{a_1 + a_2 + a_3 + \cdots + a_n}{n} = n</math>, so we must have <math>a_1 + a_2 + a_3 + \cdots + a_n = n^2</math>. The sum of consecutive odd numbers down to <math>1</math> is a perfect square, if you don't believe me, try drawing squares with the sum, so <math>a_1 = 1, a_2 = 3, a_3 = 5, \cdots , a_n = 2(n-1) + 1</math>, so the answer is <math>a_{2008} = 2(2007) + 1 = \boxed{\textbf{(B) 4015}}</math>.
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==Solution 3==
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Let the mean be <math>\frac{(a)+(a+d)+(a+2d)+...+(a+(n-1)) \cdot d)}{n}</math>
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<math>=\frac{n \cdot a}{n} + \frac{(1+2+3+...+(n-1)) \cdot d}{n}</math>
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<math>=a + \frac{n \cdot (n-1) \cdot d}{2n}</math>
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<math>=a+ \frac{(n-1) \cdot d}{2}</math>
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Note that this is also equal to n
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<math>a+ \frac{(n-1) \cdot d}{2}=n</math>
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<math>\therefore 2a+ (n-1) \cdot d=2n</math>
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1st term + nth term <math>=2n=2 \cdot 2008=4016</math>
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Now note that, from previous solutions, the first term is 1, hence the 2008th term is <math>4016-1=\boxed{\textbf{(B) 4015}}</math>
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~anshulb
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==Solution 4 (Using Answer Choices)==
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From inspection, we see that the sum of the sequence is <math>n^2</math>. We also notice that <math>n^2</math> is the sum of the first <math>n</math> odd integers. Because <math>4015</math> is the only odd integer, <math>\boxed{\textbf{(B) 4015}}</math> is the answer.
  
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==Solution==
 
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{{solution}}
 
  
 
==See also==
 
==See also==
 
{{AMC10 box|year=2008|ab=B|num-b=12|num-a=14}}
 
{{AMC10 box|year=2008|ab=B|num-b=12|num-a=14}}
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{{MAA Notice}}

Latest revision as of 04:10, 1 October 2023

Problem

For each positive integer $n$, the mean of the first $n$ terms of a sequence is $n$. What is the $2008^{\text{th}}$ term of the sequence?

$\mathrm{(A)}\ {{{2008}}} \qquad \mathrm{(B)}\ {{{4015}}} \qquad \mathrm{(C)}\ {{{4016}}} \qquad \mathrm{(D)}\ {{{4,030,056}}} \qquad \mathrm{(E)}\ {{{4,032,064}}}$

Solution 1

Since the mean of the first $n$ terms is $n$, the sum of the first $n$ terms is $n^2$. Thus, the sum of the first $2007$ terms is $2007^2$ and the sum of the first $2008$ terms is $2008^2$. Hence, the $2008^{\text{th}}$ term of the sequence is $2008^2-2007^2=(2008+2007)(2008-2007)=4015\Rightarrow \boxed{\textbf{(B) 4015}}$

Note that $n^2$ is the sum of the first n odd integers.


Solution 2

Let $a_1, a_2, a_3, \cdots, a_n$ be the terms of the sequence. We know $\frac{a_1 + a_2 + a_3 + \cdots + a_n}{n} = n$, so we must have $a_1 + a_2 + a_3 + \cdots + a_n = n^2$. The sum of consecutive odd numbers down to $1$ is a perfect square, if you don't believe me, try drawing squares with the sum, so $a_1 = 1, a_2 = 3, a_3 = 5, \cdots , a_n = 2(n-1) + 1$, so the answer is $a_{2008} = 2(2007) + 1 = \boxed{\textbf{(B) 4015}}$.

Solution 3

Let the mean be $\frac{(a)+(a+d)+(a+2d)+...+(a+(n-1)) \cdot d)}{n}$ $=\frac{n \cdot a}{n} + \frac{(1+2+3+...+(n-1)) \cdot d}{n}$ $=a + \frac{n \cdot (n-1) \cdot d}{2n}$ $=a+ \frac{(n-1) \cdot d}{2}$

Note that this is also equal to n

$a+ \frac{(n-1) \cdot d}{2}=n$ $\therefore 2a+ (n-1) \cdot d=2n$

1st term + nth term $=2n=2 \cdot 2008=4016$ Now note that, from previous solutions, the first term is 1, hence the 2008th term is $4016-1=\boxed{\textbf{(B) 4015}}$

~anshulb

Solution 4 (Using Answer Choices)

From inspection, we see that the sum of the sequence is $n^2$. We also notice that $n^2$ is the sum of the first $n$ odd integers. Because $4015$ is the only odd integer, $\boxed{\textbf{(B) 4015}}$ is the answer.


See also

2008 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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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