Difference between revisions of "2011 AMC 10A Problems/Problem 4"

(Created page with 'Let X and Y be the following sums of arithmetic sequences: <cmath> \begin{eqnarray*}X &=& 10+12+14+\cdots+100,\\ Y &=& 12+14+16+\cdots+102.\end[eqnarray*} </cmath> What is the…')
 
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<math> \textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112 </math>
 
<math> \textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112 </math>
  
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== <math>\mathbf{Solution}</math> ==
 
We see that both sequences have equal numbers of terms, so reformat the sequence to look like:  
 
We see that both sequences have equal numbers of terms, so reformat the sequence to look like:  
  
Y =_____12 + 14 + ... + 100 + 102
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<cmath>\begin{align*}
 
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Y = \ &12 + 14 + \cdots + 100 + 102\\
X = 10 + 12 +  14 + ... + 100
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X = 10 \ + \ &12 +  14 + \cdots + 100\\
 
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\end{align*}</cmath>
 
From here it is obvious that <math>Y - X = 102 - 10 = 92</math>
 
From here it is obvious that <math>Y - X = 102 - 10 = 92</math>

Revision as of 23:51, 15 February 2011

Let X and Y be the following sums of arithmetic sequences:

\begin{eqnarray*}X &=& 10+12+14+\cdots+100,\\ Y &=& 12+14+16+\cdots+102.\end[eqnarray*} (Error compiling LaTeX. Unknown error_msg)

What is the value of Y - X?

$\textbf{(A)}\ 92\qquad\textbf{(B)}\ 98\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 102\qquad\textbf{(E)}\ 112$

$\mathbf{Solution}$

We see that both sequences have equal numbers of terms, so reformat the sequence to look like:

\begin{align*} Y = \ &12 + 14 + \cdots + 100 + 102\\ X = 10 \ + \ &12 +  14 + \cdots + 100\\ \end{align*} From here it is obvious that $Y - X = 102 - 10 = 92$