Question

Given a positive integer n, the rational number r satisfies

2n/n - 2n/(n + 1) = r*2n/n.

Express r in terms of n. (Your expression should be as simplified as possible.)

Answer 1

To express r in terms of n, we can start by simplifying the left-hand side of the equation:

2n/n - 2n/(n + 1) = (2n*(n + 1) - 2n*n) / (n*(n + 1))
= (2n)/(n + 1)

Now we can set this equal to r*2n/n and solve for r:

(2n)/(n + 1) = r*2n/n
2/(n + 1) = r

Thus, the simplified expression for r in terms of n is:

r = 2/(n + 1).