Efficient method for computing the product of the first 8 terms of a recursive sequence

Question

The problem I am trying to solve is the following:

Let $x_1=97,$ and for $n>1,$ define $x_n=\frac{n}{x_{n-1}}.$ Calculate $x_1x_2 \cdots x_8.$

I tried the painstaking fail safe method for the manual computation of each term.

$x_2=\frac{2}{97}$, $x_3=\frac{291}{2}$, $x_4=\frac{8}{291}$, $x_5=\frac{1455}{8}$, $x_6=\frac{48}{1455}$, $x_7=\frac{10185}{48}$, and $x_8=\frac{384}{10185}$.

However, I realize that actually finding the product of these terms is highly inefficient and impractical in a math contest with no calculator.

My question is, what I can I do to solve this more timely? Is there a closed form shortcut?

Thank you.

Answer 1

Notice that $x_1·x_2 = \color {red}{2}$, $x_2·x_3 = \color {red}{3}$, and the same pattern continues to $x_7 · x_8 = \color {red}{8}$. Therefore, multiplying $$(x_1·x_2)·(x_3·x_4)·(x_5·x_6)·(x_7·x_8)$$ is equal to $2·4·6·8 = 384$ for all situations, regardless of $x_1$.

Your answer is 384.

The closed form is 384, and the same for recursive.