Difference between revisions of "Mock AIME 1 2007-2008 Problems/Problem 15"
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Revision as of 20:37, 2 April 2008
Problem
The sum
can be written in the form
, where
are trigonometric functions and
are degrees
. Find
.
Solution
By the product-to-sum identities, we know that , so
:
&= \sum_{x=2}^{44} \cos(x-1) - \cos(x+1) + \frac{1}{\cos(x+1)} - \frac{1}{\cos(x-1)}\\ &=\sum_{x=2}^{44} \frac{\cos^2(x-1)-1}{\cos(x-1)} - \frac{\cos^2(x+1)-1}{\cos(x+1)}\\
&=\sum_{x=2}^{44} \left(\frac{\sin^2(x+1)}{\cos(x+1)}\right) - \left(\frac{\sin^2(x-1)}{\cos(x-1)}\right)$ (Error compiling LaTeX. Unknown error_msg)This sum telescopes (in other words, when we expand the sum, all of the intermediate terms will cancel) to
We now have the desired four terms. There are a couple of ways to express as primitive trigonometric functions; for example, if we move a
to the denominator, we could express it as
. Either way, we have
, and the answer is
.
See also
Mock AIME 1 2007-2008 (Problems, Source) | ||
Preceded by Problem 14 |
Followed by Last problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |