Let a denote the least term, and z the greatest term, d the common difference, then the principal properties are expressed by these equations, viz. Moreover, when the first term a is 0 or nothing, the theorems become z = d (n − 1) and s = zn. EXAMPLES FOR PRACTICE. 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, 12 x 20 1 +40=41, is the last term. = × 20 21 x 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 67; required the sum of the series? First, 199 3.66 Then quired. X 67 = 100 X 67 = 6700, the sum re 3. To find the sum of 100 terms of the natural numbers 1, 2, 3, 4, 5, 6, &c. And. 5050. 4. * Required the sum of 99 terms of the odd numbers 1, 3, 5, 7, 9, &c. Ans. 9801. *The sum of any number (x) of terms of the arithmetical series of odd numbers 1, 3, 5, 7, 9, &c. is equal to the square (na) of that number. That is, If 1, 3, 5, 7, 9, &c. be the numbers, then will 1, 2, 3, 4, 52, be the sums of 1,2 3, &c. terms, Thus, 0 +1 = 1 or 12, the sum of 1 term, 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 OF ARITHMETICAL PROGRES QU. I. A TRIANGULAR Battalion * consists of thirty ranks, in which the first rank is formed of one man only, the second of 3; the 3d of 5; and so on: What is the strength of such a triangular battalion ? Answer, 900 men. QU. II. A detachment having 12 successive days to march, with orders to advance the first day only 2 leagues, the second 31, 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. QU. III. A brigade of sapperst, having carried on 15 yards of sap the first night, the second only 13 yards, and For, by the 1st 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 3d theorem, n n = n2 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 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 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: so on, decreasing 2 yards every night, till at last they car. ried 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. Qu. 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. Qu. 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, 2, 3, 5,2%, 6%, answers the conditions of the first detachment: and the progression 1, 34, 5, 7, answers the condition of the second detachment. 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 feel, 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 cubic foot of earth. 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. A triangular pile is formed by the continual laying of triangular horizontal courses of shot one above another, in such a manner, as that the sides of these courses, called rows, decrease by unity from the bottom row to the top row, which ends always in 1 shot. A square pile is formed by the continual laying of square horizontal courses of shot one above another, in such a man. ner, as that the sides of these courses decrease by unity from the bottom to the top row, which ends also in 1 shot. In the triangular and the square piles, the sides or faces being equilateral triangles, the shot contained in those faces form an arithmetical progression, having for first term unity, and for last term and number of terms, the shot contained J 41 in the bottom row; for the number of horizontal rows, or the number counted on one of the angles from the bottom to the top, is always equal to those counted on one side in the bottom: the sides or faces in either the triangular or square piles, are called arithmetical triangles; and the numbers contained in these, arc called triangular numbers: ABC, fig. 1, EFG, fig. 2, are arithmetical triangles. : The oblong pile may be conceived as formed from the square pile ABCD; to one side or face of which, as AD, a number of arithmetical triangles equal to the face have been added and the number of arithmetical triangles added to the square pile, by means of which the oblong pile is formed, is always one less than the shot in the top row; or which is the same, equal to the difference between the bottom row of the greater side and that of the lesser. QU. VI. To find the shot in the triangular pile ABCD, fig. 1, the bottom row AB consisting of 8 shot. Solution. The proposed pile consisting of 8 horizontal courses, each of which forms an equilateral triangle; that is, the shot contained in these being in an arithmetical progression, of which the first and last term, as also the number of terms, are known; it follows, that the sum of these particu. lar courses, or of the 8 progressions, will be the shot contained in the proposed pile; then QU. VII. To find the shot of the square pile EFGH, fig. 2, the bottom row EF consisting of 8 shot. Solution. The bottom row containing 8 shot, and the second only 7; that is, the rows forming the progression, 8, 7, 6, 5, 4, 3, 2, 1, in which each of the terms being the square root of the shot contained in each separate square |