367. Ex. 1. The first term of a G.P. is 1, the last 128, and the sum 255. Find the ratio. Ex. 2. What is the twelfth term of a G.P., if the first be 4, and the ratio 2, We have: larn−1 = 4 × 211=8192. Ex. 3. What is the sum of this series: 1†, 21, 33, &c., to 10 terms? a(r" — 1 ) __ 1 } ( (?) 10—1) __ 3 { (3)10 — 1 } Here r= 33 3 8= - 169.995. Ex. 4. The first term of a G.P. is 1, the last 65536, and the Find the sum of the series. ratio 4. Ex. 5. The sum of 12 terms of the series 64, 16, 4, &c., is required. Here r, .'. s=' = a(TM —1) = 64 ((4)12—1)_43(1—412) = (부) Ex. 6. Insert five geometrical means, between 7 and 448. Then the five means are: 14, 28, 56, 112, and 224. Ex. 7. Find the sum of the series To, Too, To3‰o, &c., ad infinitum. Here we have to use the expression s= be observed that the number of terms n being without limit, and r a proper fraction, the value or art, the last term of the series becomes less than any quantity that can be assigned, and, therefore, may be considered as nothing; and the value This quan tity is the limit to which the sum of the terms converges, and is the true expression of the sum of the terms of a series, continued A question, comprising this example, might be worded thus: if a body were propelled by a force which moved it of a mile in the first second of time, T of a mile in the second, Too of a mile in the third, &c., for ever, it would only move of a mile during all eternity. 368. These infinite series are applied with advantage to the reduction of circulating decimals. Ex. 1. Required the vulgar fraction which is equivalent to .3. This decimal may be represented by the geometrical series .3, .03, .003, &c.; hence 3 Ex. 2. Find the value of .32. .3 = Ex. 3. Find the value of the infinite decimal .5185. The series of fractions representing the value of the decimal are +the G.P. 1800, 10000, &c.; and the sum of the G.P. is 85 85 85 85 1- 10000 100 9900 51 85 5049+85 ...5185= ·+. 100 9900 9900 27 369. EXERCISES. 1. What would the price of a horse be, which is sold at 1 farthing for the first nail, 2 for the second, 4 for the third, &c., allowing 8 nails in each shoe? 2. Sessa, an Indian, having first discovered the game of chess, showed it to his Prince, Sheram, who was so delighted with the invention that he bid him ask what he would require as a reward for his ingenuity; upon which Sessa requested that he might be allowed one grain of wheat for the first square, two for the second, four for the third, &c., doubling continually to 64, the whole number of squares. Now, supposing a pint to contain 7680 of these grains, it is required to find what number of ships, each carrying 1000 tons burden, might be freighted with the produce, allowing 40 bushels to a ton; also what would be the value of the corn, at £1. 7s. 6d. per quarter. 3. A puts 6d. into a lottery, which being lost, he risks now 1s. 6d., this being likewise lost, he risks 4s. 6d.; now this process he repeated 11 times. How much must he win to recover all that he risked? 4. A charitable person gave alms to 10 poor people, each received twice as much as the one preceding; the tenth got £2. 4s. What did the first receive, and how much was distributed altogether? 5. A ball is discharged by a force which carries it 10 miles in the first minute, 9 miles in the second, and so on, in the ratio of for ever. What distance would it go? 6. Required the value of the circulating decimal .7. 7. Find the sum of the series, t, §, &c., ad infinitum. 8. Find the value of .956. 9. What debt can be discharged in a year by monthly payments, giving 4s. 2d. the first month, and three times as much every succeeding month? 370. An important application of G.P. is seen in the computation of compound interest (See Art. 283.). Let p represent any principal, R the rate per cent.; then the interest at the end of first year is px. and the amount at the end of that time Ꭱ 100' R) is p+= p(100+R 100 let this quantity=p'; the interest at R the end of second year is p'x. and the amount at that time 100' 100+ R = p2 (100+) = p (100 = p2 + 100 100 (1000XR) = which quantity let p" represent; the interest at R the end of the third year is p"> and the amount at that 100' 3 p (100+)3, let this quantity=p"; the interest at the end of R the fourth year is p"" and the amount then=p""+p""; p" 100 3 R 100 · (100+ R) = P(100+ R) x (100+ R) * = P (100+) *. 100 = amount of the given principal =l, and the therefore l=ar", and it follows that a= 1. What is the amount of £3,600, at 3 per cent., for 80 years, compound interest? 2. How many years must £6000 be lent, at 6 per cent. compound interest to amount to £34461. 4s.? 3. At what rate must £1000 be lent so as to amount to £1675 in 6 years? PERMUTATIONS AND COMBINATIONS. 372. If it were required to ascertain how many positions two persons, a and b, can take, with regard to their order, we should find that a may be on the right of bor on his left; thus they can take two different positions. Also taking two persons out of three, a, b, c, we should find that they may be placed, with regard to their order, in the following different ways: ab, ba, ac, ca, bc, cb. And if taken all three together, they may be arranged thus: abc, acb, bac, bca, cab, cba. These different orders in which any number of persons or things can be arranged, are called permutations. 373. The permutations of two things, a and b, are ab, ba, viz., 1 x 2 or 2. |