Thus, if the series be 1, 3, 5, 7, 9, 11, &c. 3. The last term of any increasing arithmetical series, is equal to the first term increased by the product of the common difference multiplied by the number of terms less one; but in a decreasing series, the last term is equal to the first term lessened by the said product. Thus, the 20th term of the series 1, 3, 5, 7, 9, &c, is 1 + 2 (20−1) = 1 + 2 × 19 = 1 +387 39. And the nth term of a, a-d, a-2d, a—3d, a-4d, &c, isa (n-1) x d = a− (n − 1) d. 4. The sum of all the terms in any series in arithmetical progression, is equal to half the sum of the two extremes multiplied by the number of terms. Thus, the sum of 1, 3, 5, 7, 9, &c, continued to the 10th (1 +19) x 10 20 x 10 term, is 10 x 10 = 100. 1. The first term of an increasing arithmetical series is 1, the common difference 2, and the number of terms 21; required the sum of the series? First, 1+2×201 †40 = 41, is the last term. × 2021 × 20 420, the sum required. 2. The first term of a decreasing arithmetical series is 199, the common difference 3, and the number of terms 37; required the sum of the series? First, 199-3.66 = 199—198 — 1, is the last term. ! Then quired. 199 +1. 1: 6700, the sum re 2 3. To find the sum of 100,terms of the natural numbers 1, 2, 3, 4, 5, 6, &c, VOL. I. P Ans, 5050 4. Required 4. *Required the sum of 99 terms of the odd numbers 1, 3, 5, 7, 9, &c. Ans. 9811. 5. The first term of a decreasing arithmetical series is 10, the common difference, and the number of terms 21; required the sum of the series ? Ans. 140. 6. One hundred stones being placed on the ground, in a straight line, at the distance of 2 yards from each other; how far will a person travel, who shall bring them one by one to a basket, which is placed 2 yards from the first stone? Ans. 11 miles and 840 yards. APPLICATION or ARITHMETICAL PROGRESSION TO MILITARY AFFAIRS. QUESTION I. A TRIANGULAR Battalion, consisting of thirty ranks, in which the first rank is formed of one man only, the second * The sum of any number (n) of terms of the arithmetical series of odd number 1, 3, 5, 7, 9, &c, is equal to the square (n2) of that number. That is, If 1, 3, 5, 7, 9, &c, be the numbers, then will 12, 22, 3, 4, 5%, &c, be the sums of 1,2, 3, &c, terms. 1+ 3 = 4 + 5 = 1 or 12, the sum of 1 term, For, by the 3d theorem, 1+2 (n − 1) = 1 + 2n−2 = 2n−1 is the last term, when the number of terms is n; to this last term 2n-1, add the first term 1, gives 2n the sum of the extremes, or n half the sum of the extremes; then, by the 4th theorem, nxn n is the sum of all. the terms. Hence it appears in general, that half the sum of the extremes, is always the same as the number of the terms n; and that the sum of all the terms, is the same as the square of the same number, n2. See more p. 111. on Arithmetical Proportion in the Arithmetic, By triangular battalion, is to be understood, a body of troops ranged in the form of a triangle, in which the ranks exceed each other second of 3, the third of 5; and so on: What is the strength of such a triangular battalion ? Answer, 900 men. QUESTION II. A detachment having 12 successive days to march, with orders to advance the first day only 2 leagues, the second 34, and so on, increasing 1 league each day's march: What is the length of the whole march, and what is the last day's march? Answer, the last day's march is 18 leagues, and 123 leagues is the length of the whole march. QUESTION III. 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? Answer, they were employed 7 nights, and the length of the whole sap was 63 yards. other by an equal number of men: if the first rank consist of one man only, and the difference between the ranks be also 1, then. its form is that of an equilateral triangle; and when the difference between the ranks is more than 1, its form may then be an isosceles or scalene triangle. The practice of forming troops in this order, which is now laid aside, was formerly held in greater esteem than forming them in a solid square, as admitting of a greater front, especially when the troops were to make simply a stand on all sides. * 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. QUESTION IV. 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? Answer, the difference between the ranks will be 1, and the number of gabions in the six ranks will be 39. QUESTION V. 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? The progression 7, 27, 370, 5, 6,7%, answers the conditions of the first detachment: and the progression 1, 3, 5, 7, answers the conditions of the second detachment. QUESTION VI. A deserter, in his flight, travelling at the rate of 8 leagues a day; and a detachment of dragoons being sent after him, with orders to march the first day only 2 leagues, the second 5 leagues, the third 8 leagues, and so on: What is the number of days necessary for the detachment to overtake the deserter, and what will be the number of leagues marched before he is overtaken? Answer, 5 days are necessary to overtake him; and consequently 40 leagues will be the extent of the march. * 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. QUESTION QUESTION VII. A convoy distant 35 leagues, having orders to join its camp, and to march at the rate of 5 leagues per day; its escort departing at the same time, with orders to march the first day only half a league, and the last day 9 leagues; and both the escort and convoy arriving at the same time: At what distance is the escort from the convoy at the end of each march? OF COMPUTING SHOT OR SHELLS IN A FINISHED PILE. SHOT and Shells are generally piled in three different forms, called triangular, square, or oblong piles, according as their base is either a triangle, a square, or a rectangle. Fig. 1. C G Fig. 2. E ABCD, fig. 1, is a triangular pile, E H C ABCDEF, fig. 3, is an oblong pile. *By convoy is generally meant a supply of ammunition or provisions, conveyed to a town or army. The body of men that guard this supply, is called escort. A triangular |