Difference between revisions of "1985 AIME Problems/Problem 1"
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Let <math>x_1=97</math>, and for <math>n>1</math> let<math>x_n=\frac{n}{x_{n-1}}</math>. Calculate the product <math>x_1x_2x_3x_4x_5x_6x_7x_8</math>. | Let <math>x_1=97</math>, and for <math>n>1</math> let<math>x_n=\frac{n}{x_{n-1}}</math>. Calculate the product <math>x_1x_2x_3x_4x_5x_6x_7x_8</math>. | ||
==Solution== | ==Solution== | ||
| − | {{ | + | Since <math>x_n=\frac{n}{x_{n-1}}</math>, <math>x_n \cdot x_{n - 1} = n</math>. Setting <math>n = 2, 4, 6</math> and <math>8</math> in this equation gives us respectively <math>x_1x_2 = 2</math>, <math>x_3x_4 = 4</math>, <math>x_5x_6 = 6</math> and <math>x_7x_8 = 8</math> so <math>x_1x_2x_3x_4x_5x_6x_7x_8 = 2\cdot4\cdot6\cdot8 = 384</math>. |
==See Also== | ==See Also== | ||
*[[1985 AIME Problems/Problem 2|Next Problem]] | *[[1985 AIME Problems/Problem 2|Next Problem]] | ||
| − | *[[1985 AIME]] | + | *[[1985 AIME Problems | Back to exam]] |
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
Revision as of 13:33, 18 November 2006
Problem
Let 
, and for 
let
. Calculate the product 
.
Solution
Since , 
. Setting 
and 
in this equation gives us respectively 
, 
, 
and 
so 
.
