Let x_1=97, and for n>1 let x_n=nx_n-1. Calculate the product x_1x_2… x_8.

1985 AIME Problem 1

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Let

x_1=97,

and for

n\gt1

let

x_n=\frac{n}{x_{n-1}}.

Calculate the product

x_1x_2\cdots x_8.

Answer

Multiply consecutive terms using the recurrence

Group the requested product as

(x_1x_2)(x_3x_4)(x_5x_6)(x_7x_8)

The recurrence gives

x_{n-1}x_n=n.

Therefore

\begin{aligned} x_1x_2\cdots x_8 &=(x_1x_2)(x_3x_4)\\ &\quad{}\cdot(x_5x_6)(x_7x_8)\\ &=2\cdot4\cdot6\cdot8\\ &=384. \end{aligned}