3. (Arithmetic and geometric means) Suppose that u, ,, are positive numbers The arithmetic mean and the geometric mean of these numbers are, respectively, 21 " 1/0 VA= VI and VG = Vi

(a) It is always true that VA VG Prove this inequality for n = 2.

(b) If 12, are rates of return of a stock in each of n periods, the arithmetic and geometric mean rates of return are likewise Vi and FG = (1+r) n 1/ - 1 Suppose $40 is invested During the first year it increases to $60 and during the second year it decreases to $48 What are the arithmetic and geometric mean rates of return over the 2 years?

(c) When is it appropriate to use these means to describe investment performance?