Find f(f(f(f(..f(10)..., where the above expression contains 97 fs and f(x) = 5/x

f(f(f(f(...f(10)..., where the expression is 97 f(10) is equal to 1/2.

Step-by-step explanation:

For this problem, let's simply solve a few of the entry cases and find a pattern that we can scale.  Given that f(x) = 5/x

f(x) = 5/x

f(f(x)) = 5/(5/x) = 5 * (x/5) = x

f(f(f(x))) = 5/(5/(5/x)) = 5/(5* x/5) = 5/(x) = 5/x

f(f(f(f(x = 5/(5/(5/(5/x))) = 5/(5/(5* x/5) = 5/(5/x) = 5*x/5 = x

After solving a few of the initial cases, we have found a pattern.  Notice, that when the number of functions of f are odd, the resultant is 5/x and when the number of functions of f are even, the resultant is x.  Using this pattern we can say that our functions of f(10), if the number is 97, will follow the odd pattern since 97 is an odd number.

So we can say the following:

f(f(f(f(...f(x)... = 5/x

f(f(f(f(...f(10)... = 5/(10) = 5/10 = 5 * 1 / 5 * 2 = 1/2

Hence, we have found that if we evaluate 10 inside of 97 of the function of x as defined, we will have the value 1/2.

Cheers.