387. To find the harmonic mean between two given quan. tities. Let a, b be the two quantities, H their harmonic mean; then ᄒᄒ 388. Relation between the Arithmetic, Geometric, and Harmonic Means. If A, G, H be the arithmetic, geometric, and harmonic means between a and b, we have proved that is, G is the geometric mean between A and H. Miscel 389. Miscellaneous Questions in the Progressions. laneous questions in the Progressions afford scope for much skill and ingenuity, the solution being often very neatly effected by some special artifice. find the following hints useful. The student will 1. If the same quantity be added to, or subtracted from, all the terms of an A. P., the resulting terms will form an A. P., with the same common difference as before. [Art 365.] 2. If all the terms of an A. P. be multiplied or divided by the same quantity, the resulting terms form an A. P., but with a new common difference. [Art. 365.] 3. If all the terms of a G. P. be multiplied or divided by the same quantity, the resulting terms form a G.P. with the same common ratio as before. [Art. 374.] ... 4. If a, b, c, d be in G. P., they are also in continued proportion, since by definition Conversely, a series of quantities in continued proportion may be represented by x, xr, xr2, .... Ex. 1. Find three quantities in G. P. such that their product is 343, and their sum 301. Ex. 8. The nth term of an A. P. is + terms. 2: find the sum of 49 Let a be the first term, and the last; then by putting n = 1, 49 respectively, we obtain and n = Ex. 4. If a, b, c, d, e be in G. P., prove that b+d is the geometric mean between a + c and c + e. Since a, b, c, d, e are in continued proportion, .... 1. Find the 6th term of the series 4, 2, 1}, Find the series in which 5. The 15th term is, and the 23d term is 2, 6. The 2d term is 2, and the 31st term is. 7. The 39th term is, and the 54th term is. Find the harmonic mean between 14. Insert two harmonic means between 4 and 12. 15. Insert three harmonic means between 2 and 12. 16. Insert four harmonic means between 1 and 6. 17. If G be the geometric mean between two quantities A and B, show that the ratio of the arithmetic and harmonic means of A and G is equal to the ratio of the arithmetic and harmonic means of G and B. 18. To each of three consecutive terms of a G. P., the second of the three is added. Show that the three resulting quantities are in H.P. 21. (2a + x)+ 3 a + (4 a − x) + ... to p terms. 24. If xa, y a, and z the harmonic mean between y 25. If a, b, c, d be in A. P., respectively; prove that ad a, e, f, d in G. P., a, g, h, d in H. P. ef=bh = cg. 26. If a2, b2, c2 be in A. P., prove that b+c, c+a, a+b are in H. P. CHAPTER XXXV. PERMUTATIONS AND COMBINATIONS. 390. Each of the arrangements which can be made by taking some or all of a number of things is called a permutation. Each of the groups or selections which can be made by taking some or all of a number of things is called a combination. Thus the permutations which can be made by taking the letters a, b, c, d two at a time are twelve in number; namely, ab, ac, ad, bc, bd, cd, ba, ca, da, cb, db, dc; each of these presenting a different arrangement of two letters. The combinations which can be made by taking the letters a, b, c, d two at a time are six in number; namely, ab, ac, ad, bc, bd, cd; each of these presenting a different selection of two letters. From this it appears that in forming combinations we are only concerned with the number of things each selection contains; whereas in forming permutations we have also to consider the order of the things which make up each arrangement; for instance, if from four letters a, b, c, d we make a selection of three, such as abc, this single combination admits of being arranged in the following ways: abc, acb, bca, bac, cab, cba, and so gives rise to six different permutations. |