Let $$f(x) = \frac{x^2}{x^2 - 1}.$$Find the largest integer $n$ so that $f(2) \cdot f(3) \cdot f(4) \cdots f(n-1) \cdot f(n) < 1.98.$
\(Let \;\;\;f(x) = \frac{x^2}{x^2 - 1}.\;\;\;\\\text{Find the largest integer n so that }\\ f(2) \cdot f(3) \cdot f(4) \cdots f(n-1) \cdot f(n) < 1.98.\)
(I have just written the question properly)
+130081
Note that 1.98 can be written as 1 + 98 / 100 = 198/100 = 99/50
Also note that
x^2 = x * x
______ __________
x^2 - 1 (x - 1) ( x + 1)
So we can write
2*2 3*3 4* 4 (n - 1) ( n -1) n * n
____ * _____ * _____ * ....... * ___________ * ___________ < 99 / 50
1 * 3 2 * 4 3 * 5 (n - 2) ( n) (n - 1) (n + 1)
Note that all the terms in red will be "cancelled" in the process and we will be left with
2 n < 99
____ ___ multiply both sides by (1/2) and we have that
(n + 1) 50
n < 99
______ ___
(n + 1) 100
And its obvious that the largest integer is n = 98
