1985 AIME Problems/Problem 1

Problem

Let $x_1=97$ , and for $n>1$ , let $x_n=\frac{n}{x_{n-1}}$ . Calculate the product $x_1x_2x_3x_4x_5x_6x_7x_8$ .

Solution

Since $x_n=\frac{n}{x_{n-1}}$ , $x_n \cdot x_{n - 1} = n$ . Setting $n = 2, 4, 6$ and $8$ in this equation gives us respectively $x_1x_2 = 2$ , $x_3x_4 = 4$ , $x_5x_6 = 6$ and $x_7x_8 = 8$ so $x_1x_2x_3x_4x_5x_6x_7x_8 = 2\cdot4\cdot6\cdot8 = \boxed{384}.$ Notice that the value of $x_1$ was completely unneeded!

1985 AIME (Problems • Answer Key • Resources)
Preceded by First Question	Followed by Problem 2
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1985 AIME Problems/Problem 1

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