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Difference between revisions of "2002 AMC 10B Problems/Problem 19"

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We should realize that the two equations are 100 terms apart, so by subtracting the two equations in a form like...
 
We should realize that the two equations are 100 terms apart, so by subtracting the two equations in a form like...
  
 +
<cmath>(a_{101} - a_1) + (a_{102} - a_2) +... + (a_{200} - a_{100}) = 200-100 = 100 </cmath>
  
−
<math>(a_{101} - a_1) + (a_{102} - a_2) +... + (a_{200} - a_{100}) = 200-100 = 100 </math>
+
...we get the value of the common difference of every hundred terms one hundred times. So we have to divide the answer by one hundred to get ...  
−
 
 
−
 
 
−
 
 
−
...we get the value of the common difference of every hundred terms hundred times. So we have to divide the answer by hundred to get ...  
 
  
 
<math>\frac{100}{100} = 1 </math>
 
<math>\frac{100}{100} = 1 </math>
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...the common difference of every hundred terms. Then we have to simply divide the answer by hundred again to find the common difference between one term, therefore...
 
...the common difference of every hundred terms. Then we have to simply divide the answer by hundred again to find the common difference between one term, therefore...
  
−
<math>\frac{1}{100} =\boxed{(\text{C})0.01}</math>
+
<math>\frac{1}{100} =\boxed{(\textbf{C})\ 0.01}</math>
  
 
== Solution 2 ==
 
== Solution 2 ==
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Now, let the common difference be <math> d </math>. Notice that <math> a_2-a_1=d </math>, so we merely need to find <math> d </math> to get the answer. The formula for an arithmetic sum is  
 
Now, let the common difference be <math> d </math>. Notice that <math> a_2-a_1=d </math>, so we merely need to find <math> d </math> to get the answer. The formula for an arithmetic sum is  
  
−
<math> \frac{n}{2}(2a_1+d(n-1)) </math>,
+
<math> \frac{n}{2}\left(2a_1+d(n-1)\right) </math>,
  
 
where <math> a_1 </math> is the first term, <math> n </math> is the number of terms, and <math> d </math> is the common difference. Now we use this formula to find a closed form for the first given equation and the sum of the given equations. For the first equation, we have <math> n=100 </math>. Therefore, we have  
 
where <math> a_1 </math> is the first term, <math> n </math> is the number of terms, and <math> d </math> is the common difference. Now we use this formula to find a closed form for the first given equation and the sum of the given equations. For the first equation, we have <math> n=100 </math>. Therefore, we have  
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Now we have a system of equations in terms of <math> a_1 </math> and <math> d </math>.
 
Now we have a system of equations in terms of <math> a_1 </math> and <math> d </math>.
−
Subtracting '''(1)''' from '''(2)''' eliminates <math> a_1 </math>, yielding <math> 100d=1 </math>, and <math> d=a_2-a_1=\frac{1}{100}=\boxed{(\text{C}) .01} </math>.
+
Subtracting '''(1)''' from '''(2)''' eliminates <math> a_1 </math>, yielding <math> 100d=1 </math>, and <math> d=a_2-a_1=\frac{1}{100}=\boxed{(\textbf{C})\ 0.01}</math>.
  
 
== Solution 3 ==
 
== Solution 3 ==
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<math>100x=1</math>  
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<math>100x=1</math>
 +
 
 +
 
 +
<math>x=0.01=\boxed{(\textbf{C})\ 0.01}</math>
 +
 
 +
== Video Solution by OmegaLearn ==
 +
https://youtu.be/tKsYSBdeVuw?t=4410
  
 +
~ pi_is_3.14
  
−
<math>x=0.01=\boxed{(\text{C})0.01}</math>
+
==Video Solution==
 +
https://www.youtube.com/watch?v=38p1OD_ATFE  ~David
  
 
==See Also==
 
==See Also==

Latest revision as of 09:21, 22 August 2024

Problem

Suppose that $\{a_n\}$ is an arithmetic sequence with \[a_1+a_2+\cdots+a_{100}=100 \text{ and } a_{101}+a_{102}+\cdots+a_{200}=200.\] What is the value of $a_2 - a_1 ?$

$\mathrm{(A) \ } 0.0001\qquad \mathrm{(B) \ } 0.001\qquad \mathrm{(C) \ } 0.01\qquad \mathrm{(D) \ } 0.1\qquad \mathrm{(E) \ } 1$

Solution 1

We should realize that the two equations are 100 terms apart, so by subtracting the two equations in a form like...

\[(a_{101} - a_1) + (a_{102} - a_2) +... + (a_{200} - a_{100}) = 200-100 = 100\]

...we get the value of the common difference of every hundred terms one hundred times. So we have to divide the answer by one hundred to get ...

$\frac{100}{100} = 1$

...the common difference of every hundred terms. Then we have to simply divide the answer by hundred again to find the common difference between one term, therefore...

$\frac{1}{100} =\boxed{(\textbf{C})\ 0.01}$

Solution 2

Adding the two given equations together gives

$a_1+a_2+...+a_{200}=300$.

Now, let the common difference be $d$. Notice that $a_2-a_1=d$, so we merely need to find $d$ to get the answer. The formula for an arithmetic sum is

$\frac{n}{2}\left(2a_1+d(n-1)\right)$,

where $a_1$ is the first term, $n$ is the number of terms, and $d$ is the common difference. Now we use this formula to find a closed form for the first given equation and the sum of the given equations. For the first equation, we have $n=100$. Therefore, we have

$50(2a_1+99d)=100$,

or

$2a_1+99d=2$. *(1)

For the sum of the equations (shown at the beginning of the solution) we have $n=200$, so

$100(2a_1+199d)=300$

or

$2a_1+199d=3$ *(2)

Now we have a system of equations in terms of $a_1$ and $d$. Subtracting (1) from (2) eliminates $a_1$, yielding $100d=1$, and $d=a_2-a_1=\frac{1}{100}=\boxed{(\textbf{C})\ 0.01}$.

Solution 3

Subtracting the 2 given equations yields


$(a_{101}-a_1)+(a_{102}-a_2)+(a_{103}-a_3)+...+(a_{200}-a_{100})=100$


Now express each $a_n$ in terms of first term $a_1$ and common difference $x$ between consecutive terms


$((a_1+100x)-(a_1))+((a_1+101x)-(a_1+x))+((a_1+102x)-(a_1+2x))+...+((a_1+199x)-(a_1+99x))=100$


Simplifying and canceling $a_1$ and $x$ terms gives


$100x+100x+100x+...+100x=100$


$100x\times100=100$


$100x=1$


$x=0.01=\boxed{(\textbf{C})\ 0.01}$

Video Solution by OmegaLearn

https://youtu.be/tKsYSBdeVuw?t=4410

~ pi_is_3.14

Video Solution

https://www.youtube.com/watch?v=38p1OD_ATFE ~David

See Also

2002 AMC 10B (Problems • Answer Key • Resources)
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Problem 20
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