18th Putnam 1958 Problem A6 Assume that the interest rate is r, so that capital of k becomes k(1 + r) n after n years. How much do we need to invest to be able to withdraw 1 at the end of year 1, 4 at the end of year 2, 9 at the end of year 3, 16 at the end of year 4 and so on (in perpetuity)?
To meet the withdrawal at the end of year n we need to invest n 2 /(1+r) n . Thus the total investment required is 1/(1+r) + 4/(1+r) 2 + 9/(1+r) 3 + 16/(1+r) 4 + ... . Now we know that 1 + x + x 2 + ... = 1/(1 - x). Differentiating, we get: x/(1 - x) 2 = x + 2x 2 + 3x 3 + ... . Differentiating again: x(1 + x)/(1 - x) 3 = 1 2 x + 2 2 x 2 + 3 2 x 3 + ... . Substituting x = 1/(1+r) gives (1 + r)(2 + r)/r 3 as the required investment. 18th Putnam 1958 © John Scholes jscholes@kalva.demon.co.uk 25 Feb 2002