the first day only 2 leagues, the second 34, and so on, increasing 1 league each day's march: find the length of the whole march, and the last day's march. Ans. the last day's march is 18 leagues, and the whole march, 123 leagues. 20. A brigade of sappers * having carried on 15 yards of sap the first night, the second only 13 yards, and so on, decreasing 2 yards every night, till at last they carried on in one night only 3 yards: what is the number of nights they were employed; and what is the whole length of the sap? Ans. they were employed 7 nights, and the whole sap was 63 yards. 21. A number of gabions † being given to be placed in six ranks, one above the other, in such a manner as that each rank exceeding one another equally, the first may consist of 4 gabions, and the last of 9: what is the number of gabions in the six ranks; and what is the difference between each rank? Ans. the difference between the ranks is 1, and the number of gabions is 39. 22. Two detachments, distant from each other 37 leagues, and both designing to occupy an advantageous post equi-distant from each other's camp, set out at different times; the first detachment increasing every day's march 1 league and a half, and the second detachment increasing each day's march 2 leagues: both the detachments arrive at the same time; the first after 5 days' march, and the second after 4 days' march? What is the number of leagues marched by each detachment each day? .... Ans. the first marches, 2%, 3%, 5%, 6%, leagues on the successive days, and the second 1§, 3§, 5§, 7§, leagues. 23. A triangular course of shot of n in each side is composed of n, n—1, n—2, 3, 2, 1 shot in succession. Find the number of shot, c,, in the whole n(n+1) Ans. C2 = 2 course. 24. A rectangular course of shot has n shot in its shorter side, and m+n in its longer one. How many shot are there in the course? Ans. c (m+n)n. GEOMETRICAL PROGRESSION, OR PROGRESSION BY RATIO. Ir a series of terms (three at least) be so taken that each is the product of the * A brigade of sappers consists generally of 8 men, divided equally into two parties. While one of these parties is advancing the sap, the other is furnishing the gabions, fascines, and other necessary implements: and when the first party is tired, the second takes its place, and so on, till each man in turn has been at the head of the sap. A sap is a small ditch, between 3 and 4 feet in breadth and depth; and is distinguished from the trench by its breadth only, the trench having between 10 and 15 feet breadth. As an encouragement to sappers, the pay for all the work carried on by the whole brigade is given to the survivors. Gabions are baskets, open at both ends, made of ozier twigs, and of a cylindrical form: those made use of at the trenches are 2 feet wide, and about 3 feet high; which, being filled with earth, serve as a shelter from the enemy's fire: and those made use of to construct batteries are generally higher and broader. There is another sort of gabion, made use of to raise a low parapet: its height is from 1 to 2 feet, and 1 foot wide at top, but somewhat less at bottom, to give room for placing the muzzle of a firelock between them: these gabions serve instead of sand bags. A sand bag is generally made to contain about a cubical foot of earth. preceding one of the series by some constant factor *, these terms are said to form a geometrical series. Thus, a, ar, ar2, ar3, .... ar"-1 form a series of terms in geometrical progression; as likewise do 2, 6, 18, 54, .... and 2, —6, 18, -54, .... of which the constant factors are respectively r, 3, and -3. These factors are called the ratios of the series. The following notation is generally employed : a for the first, and z for the last term; r for the common multiplier or ratio; n for the number of terms; and s or s, for the sum of n terms of the series. • The numerical or algebraical character of this factor is of no consequence; as it may be positive, negative, or imaginary, integer, fractional, or irrational. Its constancy throughout the series is the only condition essential to it. The only general method of finding » is by logarithms: but when n is known, r may be found by the solution of the equations 13, 14. When logarithms are used, equation (14) is a convenient form for n. When the series runs out ad infinitum, if the ratio be greater than 1, the sum is necessarily infinite, as r" will become infinitely great. When r = 1, the sum 0 of the series takes an indeterminate form $1 = .a: but other obvious considerations show that it is infinite. When r is less than 1, continually de 1 creases as the value of n increases, till r = 0. In this case s1 = The doctrine of geometrical progression finds its application in almost every department of mathematical inquiry; but one of the most elementary and most frequently required cases, is that of CIRCULATING DECIMALS. Ir has already been seen in the arithmetic, p. 60, that when certain vulgar fractions are converted into decimals, the terms of that decimal form a series of circulating periods of figures, always recurring in the same order. If we convert these periods into separate terms with the proper denominators indicated by their places in the decimal scale, we shall find them to constitute a geometrical series. which, again, may be conveniently written in the following manner :3('1+·12+·13+....) and 215(·13+·16+·19+.....). Employing the formula (15) for a decreasing infinite series, = for the first decimal .333 3.1.10 = 10-1 3 ..... = for the second ⚫215215 .... To take a general view of the subject, let us suppose the decimal is composed of a series of circulating periods preceded by a finite decimal. Let the circulating period be composed of n figures, which, taken integrally, denote the number N, and the preceding or finite part of the decimal be composed of m figures, which, together with the integers, all taken integrally, denote the number M. Then the entire decimal will be represented by {M+ 99 (n ... 1) places S When m = 0, '1" 1, and the circulating period commences at the decimal N }= M N 10" + EXAMPLES IN GEOMETRICAL PROGRESSION. 1. The first term is 1, the ratio 2, and the number of terms 12: find the last term and sum of the series. 3. Required the sum of 12 terms of the series, 1, 3, 9, 27, 81, ...; and of 7 terms of the series 1 3+9 27 + .... Ans. 265720, and . . . 4. Required the sum of 12 terms of the series, 1, 1, §, 27, 5, ... 5. Given r = 2, n = 6, s 189. Required a and z. 265720 ....; and 10 of Ans. 9, and . . . Ans. a 3, z = 96. Required r and s. 15624. 7. Find the tenth terms and the sum of the first ten in each of the following and find the 11th and 12th terms of each series; and then sum each series to 6 and to 7 terms. 8. Find the sum of ten terms, and the difference between that and the sum to infinity of the series + + . . . and likewise of — 3 + 1 — } + to six terms, and to infinity. Also of 8 + 4 - 2 + 1 9. Find the values of 333 ... ad inf.; and of 25 +252 +253 + I to 5 terms; and assign the 9th and 10th terms. Likewise the sum of both to infinity. 10. What is the sum of the infinite series ·12 + ·1a — ·16 + ·18 . 13. Suppose a body to move for ever in this manner, viz. 20 miles the first minute, 19 miles the second, 18 05 the third, and so on in geometrical progression: required the utmost distance it can reach. 14. Suppose the elastic power of a ball which falls from a height of 100 feet to be such as to cause it to rise 9375 of the height from which it fell; and to continue in this way diminishing the height to which it will rise in geometrical progression, till it comes to rest: how far will it have moved? GEOMETRICAL PROPORTION. If there be four quantities, a, b, c, d, such that ad = bc, they are said to be geometrical proportionals. The statement is usually written ab::c:d, the factors of one side of the equation being taken as the extreme, and those of the other side as the mean terms. In this a and c are called the antecedents, and b and d the consequents of the ratios a: b and c : d. Homologous terms are either both the antecedents or both the consequents. With this understanding we have at once the four following forms of the statement : Again, by division of the original or defining equation, and these two will each admit of four varieties of form analogous to those of (1, 2, 3, 4). Add bd to both sides of the defining equation: then ad + bd = bc + bd, or (a + b) d = (c + d) b; and b: d::a + b : c + d. By this and analogous processes we shall obtain Again, if we have several ranks of proportionals, the products of all their corresponding terms will be proportional quantities. Let them be, If four quantities be proportional, their nth roots are also proportional. For if ad bc, then a d= b2 c^, and a* : b :: c2: d2. a": dr........ I I m Also a d = bd, and a: b::c: đ I When the second and third terms are equal, we have abbc, and b2 = ac. .... (14) In this case b is called a mean proportional between a and c, and c a third proportional to a and b. In a geometrical series we have the terms continually proportional: viz. * Several technical terms are used by geometers to distinguish these different modes of combination; for which consult the fifth book of Euclid. |