2 MATHEMATICAL THEORY OF INVESTMENT ILLUSTRATIVE EXAMPLES. The progression 1, 2, 4, 8, 16 is an increasing geometrical progression with ratio 2, while the progression 1' t> I' ^ A is a decreasing geometrical progression with ratio ^. When a progression is given, the student should be able to determine quickly whether it is arithmetical or geometrical or whether it belongs to some other class. The test is supplied by the respective definitions. A progression cannot be at the same time arithmetical and geo- metrical, except in the trivial case where the common difference is zero or the ratio is 1. For example, the succession 2, 2, 2, 2 may be considered as an arithmetical progression with common difference zero or as a geometrical progression with ratio 1 (see Examples 12 and 13, § 2). The important things to be considered in connection with pro- gressions are the first term, the last term, the number of terms, the common difference if the progression be arithmetical or the ratio if it be geometrical, and the sum of all the terms. It will soon appear that any one of these five numbers may be expressed in terms of three others. In other words, if three of the five num- bers are given, it is possible to find out all about the progression, provided it is known whether it is arithmetical or geometrical. Progressions play a very important part in the theory of investment. 2. Derivation and use of formulas. The letters a, I, n, d, and s are used to denote the first term, the last term, the number of terms, the common difference, and the sum of all the terms of an arithmetical progression. The terms are then a, a+d, a+2d, a+3d, ,a+(n—l')d, and the last, or nth, term is clearly given by the formula ; = a + (n — 1) d. (A) PROGRESSIONS 3 The sum of all the terms is s=a+a+d+a+2d+ + a+ (n—V)d, or, if the terms be written in reverse order, g = l+l --d +l-2d ++1- (n-l')d. If these two identities be added, the result will be 2 s = (a +1) + (a + 0 + to n terms; whence s = . (a + I). (5) Similarly, the first term, the last term, the number of terms, the ratio, and the sum of all the terms of a geometrical progression are denoted by the letters a, I, n, r, and s. The terms of the series are then a, ar, ar2, ,ar"-1, and the formula for the last, or nth, term is l=ar"-1. (C) The sum of n terms written out at length is s=a+ar+ar'+---+ ar"~1. To obtain a formula for computing the sum, this equation is first multiplied by r. The resulting equation is rs = ar + ar1 + + ar""1 + ar". If now the equation for s be subtracted from the equation giving the value of rs, the result is rs — s = ar" — a, and the value of 8 obtained from this equation is ^a7^a=a-^. (7)) r—1 1—r The expression ar" is equivalent to ar'-1 r, and by equation (C) ar""1 r = rl. 4 MATHEMATICAL THEOEY OE INVESTMENT If ar" be replaced by its value rl, equation (Z>) takes the con- venient form .=^——^. (^) r—1 ' ' By means of formulas (A') and (J?), considered as a pair of simultaneous equations, any possible problem in arithmetical progressions may be solved when three of the numbers a, ?, n, d, and 8 are given. The resulting equations, when numbers are substi- tuted for the three known quantities, will constitute either a linear system or a linear quadratic system, and a solution can be found by elementary algebra in every case. In a similar fashion, problems in geometrical progressions may be solved by means of equations ((7) and (D) or (-D'). In certain cases, however, the simultaneous system cannot be solved by elementary means. This is always true when n is unknown, and usually true when r is unknown. Methods for obtaining a solution when n is unknown will be given in § 15. The arithmetical mean of two numbers is denned to be one half their sum. It is easy to see that if three numbers are in arith- metical progression, the second is the arithmetical mean of the first and the third, for if a, b, and <? are in arithmetical progression, b = a + d and c = a + 2 d; ,,, a+c a+a+2d 2(a+d) , so that -^-=——^——= 2 -b- For example, 6^ is the arithmetical mean of 5 and 8 and is the middle term of the arithmetical progression 5, 6^, 8. The geometrical mean of two numbers is the square root of their product. If three numbers are in geometrical progression, the second is the geometrical mean of the first and the third, for if a, b, and c are in geometrical progression, then b = ar and c = ar2. Consequently, Vac = Va ar2 = ar = b. PROGRESSIONS . 5 As an example, 6 is the geometrical mean of the numbers 3 and 12. Likewise Vl5 is the geometrical mean of the numbers 3 and 5. In the solution of the examples the student should pay par- ticular attention to the setting up of the pair of simultaneous equations from which the solution is obtained. It is advisable for the beginner to write down both equations, even though only one is needed. EXAMPLES 1. The sum of a number of terms of a progression whose first three terms are 32, 24, and 16 is 80. How many terms has the progression ? Solution.The progression is arithmetical with the common difference d=—S. Furthermore, a = 32, s = 80. If these values for a, d, and s are inserted in equations (A) and (B), the equations become ;=32-(n-l)8, 80=|(32+0. Eliminating ;, which is not called for, we find the quadratic equation n2 — 9 n + 20 = 0, whose solution is given by n = 4 and n = 5. It is easy to see that the sum of four terms is the same as the sum of five terms, since the fifth term is 0. 2. The sum of eight terms of the progression of which three terms are 8, 16, 32 is 510. What is. the first term ? Solution. In this problem the series is geometrical and n = 8, r = 2, s = 610. These values, substituted in equatious (C) and (D), give (= a 27 .,..1^. In this case the value of the required number a may be found from the second equation alone, but if the equations (C) and (D'} had been used, it would have been necessary to determine a by means of the two equations ;=a27, 2;-a s = ——— '