# 2008 AMC 12A Problems/Problem 4

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The following problem is from both the 2008 AMC 12A #4 and 2008 AMC 10A #5, so both problems redirect to this page.

## Problem

Which of the following is equal to the product $$\frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12}\cdot\cdots\cdot\frac{4n+4}{4n}\cdot\cdots\cdot\frac{2008}{2004}?$$ $\textbf{(A)}\ 251\qquad\textbf{(B)}\ 502\qquad\textbf{(C)}\ 1004\qquad\textbf{(D)}\ 2008\qquad\textbf{(E)}\ 4016$

## Solution 1 $\frac {8}{4}\cdot\frac {12}{8}\cdot\frac {16}{12}\cdots\frac {4n + 4}{4n}\cdots\frac {2008}{2004} = \frac {1}{4}\cdot\left(\frac {8}{8}\cdot\frac {12}{12}\cdots\frac {4n}{4n}\cdots\frac {2004}{2004}\right)\cdot 2008 = \frac{2008}{4} =$ $502 \Rightarrow B$.

## Solution 2

Notice that everything cancels out except for $2008$ in the numerator and $4$ in the denominator.

Thus, the product is $\frac{2008}{4}=502$, and the answer is $\textbf{(B)}$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 