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> | ||
| − | + | == <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 = | + | <cmath>\begin{align*} |
| − | + | Y = \ &12 + 14 + \cdots + 100 + 102\\ | |
| − | X = 10 + 12 + 14 + | + | X = 10 \ + \ &12 + 14 + \cdots + 100\\ |
| − | + | \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 22:51, 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?
We see that both sequences have equal numbers of terms, so reformat the sequence to look like:
From here it is obvious that 
