Difference between revisions of "2008 AMC 12A Problems/Problem 4"
m (wikified) |
I like pie (talk | contribs) (Formatting of problem, second solution) |
||
| Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
Which of the following is equal to the [[product]] | Which of the following is equal to the [[product]] | ||
| + | <cmath>\frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12}\cdot\cdots\cdot\frac{4n+4}{4n}\cdot\cdots\cdot\frac{2008}{2004}?</cmath> | ||
| − | + | <math>\mathrm{(A)}\ 251\qquad\mathrm{(B)}\ 502\qquad\mathrm{(C)}\ 1004\qquad\mathrm{(D)}\ 2008\qquad\mathrm{(E)}\ 4016</math> | |
| − | |||
| − | |||
| − | |||
| − | <math>\ | ||
==Solution== | ==Solution== | ||
| + | ===Solution 1=== | ||
<math>\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} =</math> <math>502 \Rightarrow B</math>. | <math>\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} =</math> <math>502 \Rightarrow B</math>. | ||
| + | |||
| + | ===Solution 2=== | ||
| + | Notice that everything cancels out except for <math>2008</math> in the numerator and <math>4</math> in the denominator. | ||
| + | |||
| + | Thus, the product is <math>\frac{2008}{4}=502</math>, and the answer is <math>\mathrm{(B)}</math>. | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2008|ab=A|num-b=3|num-a=5}} | {{AMC12 box|year=2008|ab=A|num-b=3|num-a=5}} | ||
Revision as of 23:38, 25 April 2008
Problem
Which of the following is equal to the product
![\[\frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12}\cdot\cdots\cdot\frac{4n+4}{4n}\cdot\cdots\cdot\frac{2008}{2004}?\]](http://latex.artofproblemsolving.com/6/1/d/61d211dd54e4b681761aa2c6c2fea286d329138e.png)
Solution
Solution 1
.
Solution 2
Notice that everything cancels out except for 
in the numerator and 
in the denominator.
Thus, the product is 
, and the answer is 
.
See Also
| 2008 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 3 |
Followed by Problem 5 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
