ties 1 1 1 1 A'B'C' D' Arithmetical Progression. And the same proof may be extended to any number of terms. have a common difference; that is, they are in be Hence every series of quantities in Harmonical Progression may easily converted into an Arithmetical Progression, and then the rules of Arithmetical Progression may be applied to it. Thus, Ex. Given a and b the first two terms of an Harmonic Series, to find the nth term. 294. The two extremes, and the number of terms, in an Harmonical Progression being given, the means, or intervening terms, may be found. Let a and I be the extremes, and n the number of terms; then since the reciprocals of the terms are in Arithmetic Progression, let b be their common difference, and being the nth term of the Arithmetic Progression, we have 171 .. the Harmonic Means are the reciprocals of these quantities, viz. COR. A single Harmonic mean between a and b, called the Harmonic Mean, will be thus, or Let a, a, b be in Harmonic Progression, then by Def. a: b :: a − x : x − b ; OBS. There is no general expression for the Sum of an Harmonic Series, since the sum of any number of quantities is not deducible from the sum of their reciprocals. 295. Series are sometimes proposed for summation which are not actually composed of terms in Arithmetical or Geometrical Progression, but which may be made to depend upon the rules of one or both by arrangement or artifice. Thus, Ex. 1. Let the sum of n terms of the following series be required, 1+5+13 +29 +61 + &c. Ex. 2. To find the sum of n terms of the series Let A1, A2, A3,...A, represent the several terms in order of the series a, a+b, a +2b, &c. and S the sum required; then AA(A,+b)3 - A1 = 3 Ab+ 3 A, b2 + b3, A+ A3 (A + A + ... + A) b + 3 (A, + A + ... + A2) b2 + nb3, and so on, when any other values are given to a and b. PERMUTATIONS AND COMBINATIONS. 296. DEF. The different orders in which any quantities can be arranged are called their Permutations. Thus the permutations of a, b, c, taken two and two together, are ab, ba, ac, ca, bc, cb; taken three and three together are abc, acb, bac, bca, cab, cba. 297. DEF. The Combinations of quantities are the different collections that can be formed out of them, without regarding the order in which the quantities are placed. Thus ab, ac, bc, are the combinations of the quantities, a, b, c, taken two and two; ab and ba, though different permutations, forming the same combination. 298. The number of permutations that can be formed out of n quantities, taken two and two together, is n (n − 1); taken three and three together, is n (n − 1) (n − 2). In n things, a, b, c, d, &c. a may be placed before each of the rest, and thus form n - 1 permutations; in the same manner, there are n 1 permutations in which b stands first; and so of the rest; therefore there are, upon the whole, n (n - 1) permutations of this kind, ab, ba, ac, ca, &c. Again, of n 1 things b, c, d, &c. taken two and two together, there are (n - 1) (n − 2) permutations, by the former part of the article, and by prefixing a to each of these, there are (n − 1) (n − 2) permutations, taken three and three, in which a stands first; the same may be said of b, c, d, &c. therefore there are, upon the whole, n (n − 1) (n − 2) such permutations. Ex. The number of Permutations of 7 things taken three together =7×6×5=210. 299. To find the number of permutations of n things taken r together. By Art. 298, the number taken two together = n (n-1) Now, suppose the law, which is here perceived, to hold generally, that is, let the number of permutations of n things a, b, c, d, &c. taken r-1 together be Then omitting a, it is equally true that the number of permutations of n-1 things b, c, d, &c. taken r-1 together is, (putting n − 1 for n), Prefix a to each of these last permutations, and there will be a set of permutations of n things taken r together in which a stands first in every permutation, the number of them being (n − 1) (n − 2) .............. (n − r + 1). The same number may be made of similar permutations in which b stands first; and so also for each of the n quantities a, b, c, d, &c. Hence the whole number of permutations which can be made of n things taken r together is if it be true that the number of permutations of n things taken r– - 1 together is That is, if the assumed law be true for any value of r, it is proved true for the next higher value. But it has been shewn to hold (Art. 298) when r = 2, and 3; therefore, it is true when r=4; and, if for 4, for 5; if for 5, for 6; and so on generally for any number. COR. The number of permutations of n things taken all together is n(n-1) (n-2)...... (n − n + 1) Ex. 1. Required the number of different ways in which 6 persons can be arranged at a dinner table. Number required = number of permutations of 6 things taken all together 6 × 5 × 4 × 3 × 2 × 1 = 720. Ex. 2. Required the number of changes which can be rung upon 12 bells. Number required = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 479001600. |