Given n > 8, let a = √n and b = √(n+1). Which is greater ab or ba?
Answer: ab is greater.
ab = eb ln a and ba = ea ln b. So we have to decide which of b ln a and a ln b is greater, or, equivalently, which of (ln a)/a and (ln b)/b is greater. The latter is clearly more promising. So set f(x) = (ln x)/x. Then f '(x) = 1/x2 - (ln x)/x2 which is negative for x > e. Obviously b > a, so provided a > e, (ln a)/a > (ln b)/b and hence b ln a > a ln b and ab > ba. But e2 < 9, so the result is certainly true for n ≥ 9.
3rd Putnam 1940
© John Scholes
15 Sep 1999