In a similar manner the sum of n terms may be found. Find the sum of n terms, and of an infinite number of terms of each of the following series: (8.) Since 151, 23=3+5, 357+9+11, 45 13 +15+17+ 19, &c., write down n3 according to the same law, and verify the result. Ans. n3 (no-n+1)+(n2—n+3).. +(n2+n−1)=n3. 89. When cannon balls are piled in the form of a pyramid, the base may be either a square or an equilateral triangle. In the former case, if n denote the number of balls in one side of the base, the numbers in the successive layers will be n%, (n−1), (n−2), &c. terminating in a single ball at the top. .. total number of balls = 1o +2o+3o+ ... or (§ 87. Exam. 2.) = }n (n+1) (2n+1). n2, 90. If the base be an equilateral triangle, and if n denote the number of balls in one of its sides, the number in the bottom layer will be 1+2+3+ ... + n = {n(n+1) (Exer. 5., p. 80.) ... By taking n successively equal to 1, 2, 3, &c., in the expression n (n+1), we find the number of balls in the several layers, commencing at the top, to be .. total number of balls = {1.2+2.3+3.4+..n(n+)}, or (Exer. 8. p. 80.) = ‡n(n+1)(n+2). 91. In addition to these two kinds of piles, another is sometimes used, in which the base is a rectangle with unequal sides. In this case the pile will terminate in a single row, containing one ball more than the difference of the numbers in the greater and less sides of the base. Hence, if d denote this difference, the numbers in the several layers will be d+1, 2(d+2), 3(d+3), &c., and in general n(d+n). .. total number of balls (d+1)+2(d+2)+...n(d+n) (§ 87. Exam. 2.) and (Ex. 7. p. 68.) (1.) Find the number of balls in a square pile, the num ber in one side of the base being 60. Ans. 73810. (2.) Find the number in a triangular pile, with 60 in one side of the base. Ans. 37820. (3.) Find the number in a rectangular pile, the numbers in the greater and less sides of the base being 70 and 50 respectively. Ans. 68425. (4.) Find the numbers in two incomplete piles, the one square and the other triangular; each having 20 layers, and 20 balls in the side of the uppermost layer. Ans. 18070, and 9330. (5.) Find the number in an incomplete rectangular pile of 20 layers, having 20 balls in the less side, and 30 in the greater of the uppermost layer. Ans. 23970. CHAPTER XIII. MISCELLANEOUS EXERCISES. (1.) If A and B can perform a piece of work in c days; A and C in 6 days; and B and C in a days; in how many days will each alone perform it? and how long will they take all working together? (2.) Find the time between 4 and 5 o'clock, when the hour and minute hands of a watch are exactly together. Ans. 21 minutes past 4. (3.) Find the time after h o'clock, when the hour and minute hands are distant m of the minute divisions from each other. Ans. (5hm). (4.) Divide a into three parts, such that if the first be divided by m, the second by n, and the third by p, the quotients shall be all equal. (5.) A and B set out at the same time from two places 333 miles distant, intending to meet on the road; A travels at the rate of 7 miles in 2 hours, and B 8 miles in 3 hours: in what time will they meet, and how far will each then have travelled? Ans. 54 hours; A 189 miles, and B 144 miles. (6.) A allows B a start of 2 hours, and then sets out in pursuit of him. B travels uniformly at a certain rate; but after A has travelled 3 hours, he discovers that B is travelling 24 miles per hour faster than himself; A therefore now doubles his speed, and overtakes B in 4 hours more: find their rates of travelling, and the distance travelled. Ans. A's rate at first 11 miles per hour; B's, 13; and the distance 123 miles. } (7.) A hare is 50 of its own leaps before a greyhound, and takes 3 leaps for every 2 of the greyhound; but the latter passes over as much ground in 1 leap as the former in 2: how many leaps will each take before the hare is caught? Ans. The greyhound 100, and the hare 150. (8.) Bought m+n sheep at s shillings per head; sold m of them at a profit of 10 per cent., and the rest n at a profit of 20 per cent.: how much is gained on the whole, and how much per cent. ? 8 Ans. (m+2n) shillings; and 10m+2n 10 m + n per cent. (9.) Find a number consisting of two places of figures which is equal to 4 times the sum of its digits; but if 6 be added to it, it will then be equal to 7 times the unit figure. Ans. 36. (10.) A person has two casks, each containing a certain quantity of liquid; he pours from the first into the second as much as the second contained at first; then from the second into the first as much as was left in the first; and, lastly, from the first into the second as much as was left in the second; there are now exactly 24 gallons in each: how much did each contain at first? Ans. 33 gallons, and 15. (11.) Find two numbers in the ratio of m to n, whose dif ference is equal to their product. Ans. mn m-n and m b m + 1. (13.) From a cask of wine containing a gallons, 6 gallons are drawn off, and the cask filled up with water; find how much wine remains in the cask after this has been repeated n times. Ans. (a - b)n an-1 (14.) The number of permutations of n things taken r together 5 times the number taken (r-1) together; and the number of combinations taken r together the number taken (1) together; find n and r. Ans. n 9; and r = 5. (15.) Expand (a3 — 23) into a series, both by the binomial theorem and by the method of indeterminate coeffi (16.) Find the sum of n terms, and of an infinite number of terms of the series: 1 . Ans. sn = § ±(n + 1) (n + 2); % = §• (17.) Find two numbers whose product 45; and the difference of their squares: the square of their difference :: 7:2. Ans. 9, and 5. (18.) Prove the rules for Addition and Subtraction. (See § 14., 15., 16., and 17.) (19.) Prove that in Multiplication and Division like signs give +, and unlike signs give. (§ 18.) (20.) Prove the rule for finding the greatest common measure of two quantities. (§ 30.) |