168 MATHEMATICAL THEORY OF INVESTMENT $2.012, and if the annuity were payable annually, the second mem- ber would be $202.44, so that we are reasonably sure that i lies between .05 and .06. Let i = .055 + h. We know that h is a small number, certainly less than .01, so that the binomial expansions for (1.055 + h')^"- and (1.055 + A)12 will converge rapidly. Equation (3) takes the form ^, (1.055+7012^^ ^ (1.055 + A)"-! Expanding the powers of the binomials in (4) and dropping powers of h above the first, we find 2oo = (1-055)^ + -^- (1.055)^ -1 (1.055)" + ^(1.055)~ H h -1 1.90971+21.8726^-1 -T00447+.07934A-1 ' Clearing of fractions, .89400 +15.868 h = .90971 + 21.8726 h. Solving for A, h = - g^g =-.0026+. This value of h gives i = .0524 .. Starting with i = .0524, we find the second approximation in exactly the same way. EXAMPLE A man pays $2 to a building and loan association on the first of each month for 7 years, -when he receives $200 as the face value of his stock. What rate of interest does he receive? 62. The time required for stock to mature. PROBLEM. To find the approximate time in which stock will mature when the maturing value, the monthly payment, and the approximate rate of interest are known.