Difference between revisions of "AoPS Wiki talk:Problem of the Day/July 14, 2011"
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==Solution== | ==Solution== | ||
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We begin by factoring the given expression, <math>\prod_{n=1}^{39}\frac{n^2+6n+9}{n^2+6n+8}</math>, to <math>\prod_{n=1}^{39}\frac{(n+3)^2}{(n+2)(n+4)}</math>. | We begin by factoring the given expression, <math>\prod_{n=1}^{39}\frac{n^2+6n+9}{n^2+6n+8}</math>, to <math>\prod_{n=1}^{39}\frac{(n+3)^2}{(n+2)(n+4)}</math>. | ||
Then, writing this as multiple products and shifting the indices for clarity, we get <math>\frac{(\prod_{n=4}^{42}n)^2}{(\prod_{n=3}^{41}n)(\prod_{n=5}^{43}n)}</math>. | Then, writing this as multiple products and shifting the indices for clarity, we get <math>\frac{(\prod_{n=4}^{42}n)^2}{(\prod_{n=3}^{41}n)(\prod_{n=5}^{43}n)}</math>. |
Revision as of 17:14, 14 July 2011
Problem
AoPSWiki:Problem of the Day/July 14, 2011
Solution
We begin by factoring the given expression, , to . Then, writing this as multiple products and shifting the indices for clarity, we get . Clearly, this equals . At this point, all that is left is arithmetic. The expression equals . This trivially simplifies to .