2011 AMC 10A Problems/Problem 4

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 $Y - X = 102 - 10 = 92$

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2011 AMC 10A Problems/Problem 4

$\mathbf{Solution}$