Here p=0,2 = -2 ; theref. n m = a u m 4.0 3 II 3n a 6 7. Find the value of (a3 – 65) or (as_63)in a series. 8. To find the value of (as + xs) or (as +357 in a series. Ecclx, and-=-= 1 1 =A, the 1st term of the series. 1 2x - AQ =-2-X 2as I = B, the 2d term. a? a3 mn 2x 3r% BQ=-X-X =--= 3a^4x4 = C, the 3d. 2n @s - 2n 3.co - co=-x x =40533 = D. a4 a a 5 Hence aos+20*38 + 32***? + 42 533+ &c, or 1 3.ro 4x3 5.04 + -+ + &c, is the series required. Q? 23 a 4 as aa 3. To find the value of -, in an infinite series. x2 x 3 a3 I in a series. (a2 +32) 3.7* 5.36 &c. in an infinite series. 26 362 Ans. 1+-+ + + &c. 22 6. To expand Vaz-x? or (a2-82)# in a series. 2 &e. 2a 8a3 16as 12847 60 Ans. A &c. 3a% 9a5 81a8 &c. 4. To expand diantais 5. To expand 463 564 a2-62 9. To find the square root of in an infinite series. a? +62 6 x? Ans. I &c. 2a 223 23 10. Find the cube root of in a series. 1469 &c. 3a3 926 81a 266 ARITHMETICAL PROPORTION. ARITHMETICAL PROPORTION is the relation between two numbers with respect to their difference. Four quantities are in Arithmetical Proportion, when the difference between the first and second is equal to the difference, between the third and fourth. Thus, 4, 6, 7, 9, and a, 4 + d, b, 6 x d are in arithmetical proportion. Arithmetical Progression is when a series of quantities have all the same common difference, or when they either increase or decrease by the same common difference. Thus, 2, 4, 6, 8, 10, 12, &c, are in arithmetical progression, having the common difference 2 ; and a, a + d, a + 2d, a + 3d, a + 4d, a + 5d, &c, are series in arithmetical progression, the common difference being d. The most useful part of arithmetical proportion is containe in the following theorems : 1. When four quantities are in Arithmetical Proportion, the sum of the two extremes is equal to the sum of the two means. Thus, in the arithmetical 4, 6, 7, 9, the sum 4 + 9= 6 + 7 13: and in the arithmeticals a, a + d,b,6 td, the suni a + b + d = a +b+ d. 2. In any continued arithmetical progression, the sum of the two extremes is equal to the sum of any two terms at an equal distance from them. Thus, Thus, if the series be 1, 3, 5, 7, 9, 11, &c. Then I + 11 = 3 + 9 5 +7= 12. 3. The last term of any increasing arithmetical series, is cqual to the first term increased by the product of the common difference multiplied by the number of terms less one; bu: 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 x 19= 1 + 38 = 39. And the nth term of a, a-d, 2-3d, a--3d, 2 - 4d, &c, is = -(n-1) Xd=a-(n-1) d. 4. The sum of all thc 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 Perm, is = = 10 x 10 = 100. 2 2 And the sum of n terms of a, a + d, a + 2d, a + 3d, to a + md, is = (a + a + md).-=(a + 1 md) n. 2 n 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, 1 + 2 X 20 = 1 + 40 = 41, is the last term. 1+ 41 Then x 20 = 21 x 20 = 420, the sum required. 2 2. The first term of a decreasing arithmetical series 199, the common difference 3, and the number of terms 67 ; re. quired the sum of the series ? First, 199–3.66 = 199 – 198 = 1, is the last term. 199 + 1 Then X 67 = 100 X 67 = 6700, the sum re2 quired. 3 To find the sum of 100 terms of the natural numbers 1, 2, 3, 4, 5, 6, &c. Ans. 5050. 4. Required 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 ? leagues, the second 3}, 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, tili 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 imple. ments, 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 |