CHAPTER VI. RATIO AND PROGRESSION. (131.) By Ratio of two quantities, we mean their relation. When we compare quantities by seeing how much greater one is than another, we obtain arithmetical ratio. Thus, the arithmetical ratio of 6 to 4 is 2, since 6 exceeds 4 by 2; in the same way the arithmetical ratio of 11 to 7 is 4 In the relation a—c=r, r is the arithmetical ratio of a to c. (1) The first of the two terms which are compared is called the antecedent; the second is called the consequent. Thus referring to (1), we have α= antecedent. c = consequent. r= ratio. From (1), we get by transposition a=c+r, c=a-r. (2) (3) Equation (2) shows, that in an arithmetical ratio the antecedent is equal to the consequent increased by the ratio. Equation (3) in like manner shows, that the consequent is equal to the antecedent diminished by the ratio. (132.) When the arithmetical ratio of any two terms is the same as the ratio of any other two terms, the four terms together form an arithmetical proportion. Thus if a-c=r; and a'-c'=r, then will which relation is an arithmetical proportion, and is read thus: a is as much greater than c, as a' is greater than c'. Of the four quantities constituting an arithmetical proportion, the first and fourth are called the extremes, the second and third are called the means. The first and second, together, constitute the first couplet; the third and fourth constitute the second couplet. From equation (4), we get by transposing which shows, that the sum of the extremes, of an arithmetical proportion, is equal to the sum of the means. If c=a', then (4) becomes a—a'=a'—c', which changes (5) into a+c'=2a'. (6) (7) So that if three terms constitute an arithmetical proportion, the sum of the extremes will equal twice the mean. (133.) A series of quantities which increase or decrease by a constant difference form an arithmetical progression. When the series is increasing, it is called an ascending progression; when decreasing, it is called a descending progression. Thus, of the two series 1, 3, 5, 7, 9, 11, &c. 27, 23, 19, 15, 11, 7, &c. (8) (9) The first is an ascending progression, whose ratio or common difference is 2; the second is a descending progression, whose common difference is 4. (134.) If a the first term of an ascending arithmetical progression, whose common differenced, the successive terms will be a=first term, a+d second term, a+2d-third term, a+3d fourth term, (10) a+(n-1)d=nth term. If we denote the last or nth term by l, we shall have When the progression is descending, we must write -d for d in the above formulas. Suppose, in an arithmetical progression, x to be a term, which is preceded by q terms; and y to be a term which is followed by q terms; then by using (11) we have Taking the sum of (15) and (16), we get That is, the sum of any two terms equi-distant from the extremes is equal to the sum of the extremes, so that the terms will average half the sum of the extremes; consequently, the sum of all the terms equals half the sum of the extremes multiplied by the number of terms. Representing the sum of n terms by s, we have being given, the remaining two can be found, which must give rise to 20 different formulas, as given in the following table for ARITHMETICAL PROGRESSION. (135.) We have not deemed it necessary to exhibit the particular process of finding each distinct formula of the following table, since they were all derived from the two fundamental ones (11) and (18), by the usual operations of equations not exceeding the second degree. 12n, l, d= (l+a)(l—a) 2s-1-a d= |