By addition, since un+1 + Dun+1 = Un+2, we obtain Un+2=U1+(n+1)Dμ1+ ··· +("C,+”C,-1)D‚μ1+ ... r +1 ... (n+1)n(n-1)... (n+1−r+1) 1.2.3...(r-1)r Hence if the law of formation holds for un+1 it also holds for un+2, but it is true in the case of us, therefore it holds for us, and therefore universally. Hence If we take a as the first term of a given series, dy, da, d.... as the first terms of the successive orders of differences, any term of the given series is obtained from the formula 525. The Sum of n Terms of the Series. Suppose the •'• Vn+1 = 0 + nu1+ ... same as in the preceding article ; n (n − 1) Du1 + series, d1, d2, dз, · ... If, as in the preceding article, a is the first term of a given the first terms of the successive orders of differences, the sum of n terms of the given series is obtained from the formula Ex. 1. Find the 7th term and the sum of the first seven terms of the series 4, 14, 30, 52, 80, ..... Using formula, Art. 525, the sum of the first seven terms Ex. 2. Find the general term and the sum of n terms of the series 12, 40, 90, 168, 280, 432, The successive orders of difference are .... 1)(n · = 12 + 28 (n − 1)+: 22 (n-1)(n-2) 6 (n-1) (n − 2) (n − 3) 12 + = n3 + 5 n2 + 6 n. Using the formula for the sum of n terms we obtain 526. It will be seen that this method of summation will only succeed when the series is such that in forming the orders of differences we eventually come to a series in which all the terms are equal. This will always be the case if the nth term of the series is a rational integral function of n. PILES OF SHOT AND SHELLS. 527. Square Pile. To find the number of shot arranged in a complete pyramid on a square base. The top layer consists of a single shot; the next contains 4; the next 9, and so on to n2, n being the number of layers: hence the form of the series is 528. Triangular Pile. To find the number of shot arranged in a complete pyramid the base of which is an equilateral triangle. The top layer consists of a single shot; the next contains 3; the next 6; the next 10, and so on, giving a series of the form 1, 1+2, 1+2+3, 1+2+3+4,... 529. Rectangular Pile. To find the number of shot arranged in a complete pile the base of which is a rectangle. The top layer consists of a single row of shot. Suppose this row to contain m shot; then the next layer contains 2(m +1); the next 3 (m + 2), and so on, giving a series of the form m, 2m+2, 3m+6, 4m+12,... 1st order of differences m+2, m+4, 2d order of differences 3d order of differences m+6, Now let 7 and w be the number of shot in the length and width, respectively, of the base; then ml-w+1. Making these substitutions, we have 1. Find the eighth term and the sum of the first eight terms of the series 1, 8, 27, 64, 125, .... 2. Find the tenth term and the sum of the first ten terms of the series 4, 11, 28, 55, 92, .... Find the number of shot in : 3. A square pile, having 15 shot in each side of the base. 4. A triangular pile, having 18 shot in each side of the base. 5. A rectangular pile, the length and the breadth of the base containing 50 and 28 shot respectively. 6. An incomplete triangular pile, a side of the base having 25 shot, and a side of the top 14. 7. An incomplete square pile of 27 courses, having 40 shot in each side of the base. 8. Find the ninth term and the sum of the first nine terms of the series 1, 35, 7+9+ 11, .... ... The numbers 1, 2, 3,.. are often referred to as the natural numbers. 9. Find the sum of the squares of the first n natural numbers. 10. Find the sum of the cubes of the first n natural numbers. 11. The number of shot in a complete rectangular pile is 24395; if there are 34 shot in the breadth of the base, how many are there in its length? 12. The number of shot in the top layer of a square pile is 169, and in the lowest layer is 1089; how many shot does the pile contain ? 13. Find the number of shot in a complete rectangular pile of 15 courses, having 20 shot in the longer side of its base. 14. Find the number of shot in an incomplete rectangular pile, the number of shot in the sides of its upper course being 11 and 18, and the number in the shorter side of its lowest course being 30. Find the nth term and the sum of n terms of the series: 19. 30, 144, 420, 960, 1890, 3360, .... 20. What is the number of shot required to complete a rectangular pile having 15 and 6 shot in the longer and shorter side, respectively, of its upper course? 21. The number of shot in a triangular pile is greater by 150 than half the number of shot in a square pile, the number of layers in each being the same: find the number of shot in the lowest layer of the triangular pile. |