Let $$f(x) = \frac{x^2}{x^2 - 1}.$$find the largest integer $n$ so that $f(2) \cdot f(3) \cdot f(4) \cdots f(n-1) \cdot f(n) < 1.98.$

The question asks: "Let . Find the largest integer n so that f(2) · f(3) · f(4) · ... · f(n-1) · f(n) < 1.98"

The answer is n = 98

Explanation:

First thing, consider that the function can be written as:


Now, let's expand the product, substituting the function with its equation for the requested values:


As you can see, the intermediate terms cancel out with each other, leaving us with:


This is a simple inequality that can be easily solved:


200n < 198(n + 1)
200n < 198n + 198
2n < 198
n < 99

Hence, the greatest integer n < 99 (extremity excluded) is 98.