ascending; thus, 2, 6, 18, 54, 162, is an ascending series, in which 3 is the multiplier. When the multiplier is less than a unit, the series is descending; thus, 162, 54, 18, 6, 2, is a descending series, in which is the multiplier. 456. The Ratio is the constant multiplier. 45%. In every geometrical progression there are five parts to be considered, any three of which being given, the other two may be determined. They are as follows: The first term, last term, ratio, number of terms, and the sum of all the terms. The first and last terms are the extremes, and the intermediate terms are the means. CASE I. 458. To find any term, the first term, the ratio, and number of terms being given. The first term is supposed to exist independently of the ratio. Using the ratio once as a factor, we have the second term; using it twice, or its second power, we have the third term; using it three times, or its third power, we have the fourth term; and, in general, the power of the ratio in any term is one less than the number of the term. The ascending series, 2, 6, 18, 54, may be analyzed thus: 2, 2 × 3, 2 × 3 × 3, 2 × 3 × 3 × 3. In this illustration we see that 1st term, 2, is independent of the ratio. 2d term, 6 = the ratio. 2 × 3 = the first term into the 1st power of 3d term, 18 = 2 × 32 = the first term into the 2d power of the ratio. 4th term, 54 = 2 x 33 the first term into the 3d power of the ratio. RULE. Multiply the first term by that power of the ratio denoted by the number of terms less 1. EXAMPLES FOR PRACTICE. 1. The first term of a geometrical series is 4, the ratio is 3; what is the 9th term? Ans. 4 x 38 = 26244. 2. The first term is 1024, the ratio †, and the number of terms 8; what is the last term? Ans. 3. A boy bought 9 oranges, agreeing to pay 1 mill for the first orange, 2 mills for the second, and so on; what did the last orange cost him? Ans. $.256. Ans. 1607 4. The first term is 7, the ratio, and the number of terms 7; what is the last term? 5. What is the amount of $1 at compound interest for 5 years, at 7 per cent. per annum ? Ans. $1.40255+. In the above example the first term is $1, the ratio is $1.07, and the number of terms is 6. 6. A drover bought 7 oxen, agreeing to pay $3 for the first ox, $9 for the second, $27 for the third, and so on ; what did the last ox cost him? Ans. $2187. CASE II. 459. To find the sum of all the terms, the extremes and ratio being given. If we take the series 2, 8, 32, 128, 512, in which the ratio is 4, multiply each term by the ratio, and add the terms thus multiplied, we shall have = 682 = 8 +32 + 128 + 512 + 2048 = 2728 = = Four times the sum of all the terms. Once the sum of all the terms. Three times the sum of The subtraction is performed by taking the lower line or series from the upper. All the terms cancel except 2048 and 2. Taking their difference, which is 3 times the sum, and dividing by 3, the ratio less one, we must have the sum of all the terms. RULE. Multiply the greater extreme by the ratio, subtract the less extreme from the product, and divide the remainder by the ratio less 1. Let every decreasing series be inverted, and the first term called the last; then the ratio will be greater than a unit. If the series be infinite, the first term is a cipher. EXAMPLES FOR PRACTICE. 1. The first term is 2, the last term 486, and the ratio 3; what is the sum of all the terms? Ans. 728. 2. The first term is 4, the last term is 262144, and the ratio is 4; what is the sum of the series? Ans. 349524. 3. The first term of a descending series is 162, the last term 2, and the ratio ; what is the sum ? Ans. 242. 4. What is the value of 1, 2, 1, etc., to infinity? Ans. 1. In the following examples we first find the last term by the Rule under Case I. 5. What yearly debt can be discharged by monthly payments, the first being $2, the second $6, and the third $18, and so on, in geometrical progression ? Ans. $531440. 6. If a grain of wheat produce 7 grains, and these be sown the second year, each yielding the same increase, how many bushels will be produced at this rate in 12 years, if 1000 grains make a pint ? Ans. 252315 bu. 43 qt. 7. Six persons of the Morse family came to this country 200 years ago; suppose that their number has doubled every 20 years since, what would be their number now? The other cases in Progression will be found in the Higher Arithmetic. PROMISCUOUS EXAMPLES. 1. One-half the sum of two numbers is 800, and one-half the difference of the same numbers is 200; what are the numbers? 2. What number is that to which, if you add um will be 61? 3. What part of a day is 3 h. 21 min. 15 sec.? Ans. 1000 and 600. of of itself, the Ans. 55. Ans. 61 4. A commission merchant received 70 bags of wheat, each containing 3 bu. 3 pk. 3 qt.; how many bushels did he receive? 5. Four men, A, B, C, and D, are in possession of $1100; A has a certain sum, B has twice as much as A, C has $300, and D has $200 more than C, how many dollars has A? Ans. $100. 6. At a certain election, 3000 votes were cast for three candidates, A, B, and C; B had 200 more votes than A, and C had 800 more than B; how many votes were cast for A? 7. What part of 173 is 31 ? Ans. 600. 8. The difference between and of a number is 10; what is the number? Ans. 560. 9 A merchant bought a hogshead of rum for $28.35; how much water must be added to reduce the first cost to 35 cents per gallon? Ans. 18 gal. 10. A and B traded with equal sums of money; A equal to of his stock; B lost $200, and then he had how much was the original stock of each? gained a sum as much as A ; Ans. $500. 11. A farmer sold 17 bushels of barley, and 13 bushels of wheat, for $31.55; he received for the wheat 35 cents a bushel more than for the barley; what was the price of each per bushel? Ans. Barley, $.00; wheat, $1.25. 12. What is the interval of time between March 20, 21 minutes past 3 o'clock, P. M., and April 11, 5 minutes past 7 o'clock, A. M.? Ans. 21 da. 15 h. 44 min. 13. What o'clock is it when the time from noon is of the time to midnight? Ans. 5 o'clock 24 min. P. M. 14. What is the least number of gallons of wine that can be shipped in either hogsheads, tierces, or barrels, just filling the vessels, without deficit or excess? Ans. 126 gal. 15. A ferryman has four boats; one will carry 8 barrels, another 9, another 15, and another 16; what is the smallest number of barrels that will make full freight for any one, and all of the boats? 16. A and B have the same income; A saves of his, but B, by spending $30 a year more than A, at the end of four years finds himself $40 in debt; what is their income, and how much does each spend a year? Ans. (Income, $160. A spends $140. B spends $170. r 17. If a load of plaster weighing 1825 pounds cost $2.19, how much is that per ton of 2000 pounds? Ans. $2.40. 18. If 23 yards of cloth 13 yards wide cost $3.37%, what will be the cost of 36 yards 14 yards wide? Ans. $52.779. 19. I lend my neighbor $200 for 6 months; how long ought he to lend me $1000 to balance the favor? 20. Bought railroad stock to the amount of $2356.80, and found that the sum invested was 40 per cent. of what I had left; what sum had I at first? Ans. $8248.80. 21. 20 per cent. of 3 of a number is what per cent, of of it? Ans. 121. 22. Divide a prize of $10200 among 60 privates, 6 subaltern officers, 3 lieutenants, and a commander, giving to each subaltern double the share of a private, each lieutenant 3 times as much as the subaltern, and to the commander double that of a lieutenant; how much is each man's share? Ans. Com. $1200; each man, $100. 23. A is 51 miles in advance of B, who is in pursuit of him; A travels 16 miles per hour, and B 19; in how many hours will B overtake A? 24. How much wool, at 20, 30, and 54 cents per pound, must be mixed with 95 pounds at 50 cents, to make the whole mixture worth 40 cents per pound? Ans. 133 lb. at 20; 95 lb. at 30; 190 lb. at 54 cents. 25. If 240 bushels of wheat are purchased at the rate of 18 bushels for $221, and sold at the rate of 22 bushels for $33.75, what is the profit on the whole? Ans. $60. 26. My horse wagon, and harness together are worth $169; the wagon is worth 4 times the harness, and the horse is worth double the wagon; what is the value of each? Horse, $104. Ans. Wagon, $52. Harness, $ 13. 27. The shadow of a tree measures 42 feet; a length casts a shadow 18 inches at the same time; of the tree? staff 40 inches in what is the height Ans. 93 feet. 23. If a piece of land 40 rods long and 4 rods wide make an acre, how wide must it be to contain the same if it be but 25 rods long? Ans. 6 rods. 29. A, B, and C are employed to do a piece of work for $26.45; A and B together are supposed to do of it; A and C, and B and C 8, and paid proportionally; how much must each receive? 30. If 12 ounces of wool make 23 yards of cloth that is 6 quarters wide, how many pounds of wool will it take for 150 yards of cloth 4 quarters wide? 31. Six persons, A, B, C, D, E, and F, are to share among them $6300; A is to have of it, B, C, D is to have as much as A and C together, and the remainder is to be divided between E and F in the proportion of 3 to 5; how much does each one receive? 32. What is the amount of $200 for 8 years at 6 per cent. compound interest? Ans. $318.769. 33. A garrison, consisting of 360 men, was provisioned for 6 months; but at the end of 5 months they dismissed so many of the men that the remaining provision lasted 5 months longer; how many men were sent away. 34. A certain principal, at corpound interest for 5 years, at 6 per cent., will amount to $669.113: what time will the same principal amount to the same sum, at 6 per simple interest? Ans. 5 yr. 7 mo. 19.3+da. 35. Paid $148.352 for 9728 feet of pine lumber; how much was that per thousand? |