difficulty; (1) given a, r, n; (2) given a, n, 1; (3) given r, n, 1; (4) given r, n, 8. 471. Suppose, however, that a, s, n are given, and therefore are to be found. Then r would have to be found from r and the equation 8 (r−1) = a("-1); we may divide both sides by r-1, and then we shall have an equation of the (n − 1)th degree in the unknown quantity r, which therefore cannot be solved by any method yet given, if n be greater than 3. Similar remarks will hold in the case where l, s, n are given, and therefore a and r are to be found. 472. Four cases of the problem remain, namely, those four in which n is one of the quantities to be found. Suppose a, r, l given, and therefore s and n are to be found. Here n would have to be found from the equation l = ar"-1, where the unknown quantity n occurs as an exponent; nothing has been said hitherto as to the solution of such an equation. 473. To find the sum of n terms of the following series; 22. Find the sum of any number of terms in G. P. whose first and third terms are given. 23. If the common ratio of a G. P. is – 3, what is the common ratio of the series obtained by taking every fourth term of the original series? 24. The sum of £700 was divided among 4 persons, whose shares were in G. P.; and the difference between the greatest and least was to the difference between the means as 37 to 12. What were their respective shares? 25. Sum to n terms the series whose mth term is (−1)TMɑ1m. 28. A person who saved every year half as much again as he saved the previous year had in seven years saved £102. 19s. How much did he save the first year? 29. In a G. P. shew that the product of any two terms equidistant from a given term is always the same. 30. In a G. P. shew that if each term be subtracted from the succeeding, the successive differences are also in G. P. 31. The square of the arithmetical mean of two quantities is equal to the arithmetical mean of the arithmetical and geometrical means of the squares of the same two quantities. 32. In a G. P. continued to infinity, shew that each term bears a constant ratio to the sum of all that follow it. And find a series in which each term is p times the sum of all the terms that follow it. n 33. If S represent the sum of n terms of a given G. P., find the sum of S1 + S2+ Sz + 2 ...... + S. n 34. If n geometrical means be found between two quantities n a and c, their product will be (ac). ar3, ... .; 1 35. Let s denote the sum of n terms of the series a, ar, lets' denote the sum of n terms of the series a, ar ̄1, and let denote the last term of the first series; then will as = = ls'. ar-2, .; 36. If a, b, c, d be in G. P., (a2 + b2 + c2)(b2 + c2 + d2) = (ab + bc + cd)3. 37. If a, b, c, d be in G. P., 38. (ad)2 = (b−c)2 + (c− a)2 + (d—b)3. The sum of the first three terms of a G. P. = 21, and the sum of the first four terms 45; find the series. 39. Sum to n terms 12+ 32 + 52 + 72 + 41. Prove that the two quantities between which A is the arithmetical and G the geometrical mean, are given by the formula T. A. A± √√{(A + G) (A — G)}. 18 42. There are four numbers, the first three of which are in G. P., and the last three in A. P.; the sum of the first and last is 14, and the sum of the second and third is 12; find the numbers. 43. Three numbers whose sum is 15 are in A. P.; if 1, 4, and 19 be added to them respectively they are in G. P. Determine the numbers. 45. Find the sum of the infinite series a+ar + (a + ab) r2 + (a + ab + ab2) r3 + r and br being each less than unity. ... XXXII. HARMONICAL PROGRESSION. 474. Three quantities A, B, C, are said to be in Harmonical Progression when A: C: A-B: B-C. Any number of quantities are said to be in Harmonical Progression when every three consecutive quantities are in Harmonical Progression. 475. The reciprocals of quantities in Harmonical Progression are in Arithmetical Progression. Let A, B, C be in Harmonical Progression; then ACA-B: B-C, therefore = A(B-C) C(A – B). |