If the last or greatest term be not given, let it be found by Prob, 3 or 4, then subtract the least from the greatest; divide the remainder by the ratio of the progression less 1, and to the quotient add the greatest or last term. Application. Let it be required to find the sum of the following progression, 1, 3, 9, 27, 81, 243, 729 § 1 3 9 From 729 the greatest. Take Ratio 3-12)728 I the least. 27 81 243 729 1093 364 Quotient. 729 1093 QUESTIONS. 1. A man bought a horse and was to give a farthing for the first nail, 2 for the second, 4 for the third, &c. in geometrical progression. The number of nails was to be 7 in each shoe, viz. 28 nails in all: What must be paid for the horse? Answ. 2796201. 5s. 33d. 1st 0 1 2 3 4 5 1, 2, 4, 8, 16, 32, ( Prob. 3;) the last or 28th term will be found to be 134217728 Subtract Then ratio-1-1 which 13+217727 divides not, therefore add 134217728 the gr. 4)268435455 12)671088633 210)5592405 8 £279620 5 32 2. A merchant sold 15 yards of sattin; the first for 1s. the second for 2s, the third for 4s. the fourth for 8s. I demand the price of the 15 yards? Answ. 16381. 7s. 3. A draper sold 20 yards of superfine cloth; the first yard for 3d. the second for 9d, the third for 27 d. &c. in triple proportion geometrical. I demand the price of the cloth? Answ. 217924027. 10s. 4. A goldsmith sold 1b. of gold at a farthing for the first ounce, a penny for the second, 4d. for the third, &c. in quadruple proportion geometrical. 1 demand what he sold the whole for, also how much he gained by the sale thereof, supposing he gave for it 41. ounce? He sold it for 58251 8s. 54d, And gain'd 57771. Ss. 5 d. Answ. 5. A cunning servant agreed with a master (unskilled in numbers) to serve him 11 years, without any other reward for his service but the produce of 1 wheat-corn for the first year; and that product to be sow'd the second year, and so on from year to year until the end of the time, allowing the increase to be but in a tenfold proportion; that 7680 wheat corns make a pint, and is sold at 3s. bushel ? Answ. 339081. 8s. 44d. 6. A thresher work'd 20 days at a farmer's, and received for the first day's work 4 barley.corns, for the second 12 barley.corns, for the third 36 barley.corns, and so on in triple proportion geometrical. I demand what the 20 day's labour came to, supposing the pint to contain 7680 corns, and the whole quantity to be sold at 2s. 6d. bushel Answ. 1773 7s. 6d. rejecting remainders. 7. A merchant sold 30 yards of fine velvet trimmed with gold very curiously, at 2 pins for the first yard, 6 pins for the second, 18 pins for the third, &c. in triple proportion geometrical. I demand how much the velvet produced when the pins were afterwards sold at an hundred for a farthing; also, whether the said merchant gain'd or lost by the sale thereof, and how much, supposing the said velvet to have been bought at 501. yard? The velvet produced 21446992921. 13s. 0žd. The merchant gain'd 21416977921. 13s. 0 d. Answ. { СНАР. V. WH COMPOUND INTEREST. THAT Compound interest is, is already signified, which see p. 212. From which it follows, that if any sum, as 100 pound, be lent out, suppose at 5 cent. and that the interest be not paid at the year's end, there will arise a new principal of 1051. on which interest must be paid the second year, and if it runs on a third year, then the principal for the third year will be 1051, together with a year's interest of 105. i. e. 1101, 5s, &c. This being well considered, will point out a method for finding the amount of any sum for any number of years, at compound interest. As for 1. What will 5001. amount to in 3 years, at 8 cent. interest upon interest? 1. If 100: 108: 500: 540 amt. of 1st year. 100: 108: 540: 583 4s. -2d. year, 1. S. 1. S. d. 100: 108: 583 4: 629 17 1 3d year requir. Or, 1: 1.08: 500: 540 1: 1.08 :: 540 583 2 :: 583.2: 629.856. Which amounts 540,583.2. 629.856, being produced from 500 by the continual multiplication thereof by 1.08 the principal and several amounts are in a geometrical progression, (viz. 500, 540,583.2, 629.856) whose ratio is the amount of 11. for a year, viz. here 1.08, and the number of years are continually indexes of the terms. Likewise. If the amount of 11. for any number of years be multiplied by any given principal, the product will be the amount of that principal for the same time. From which consideration we draw the following Rule. Find the amount of 11. for the given time (which is to find a term in the geometrical progression from 1. whose index is the number of years given) and multiply that amount of 11. by the principal given. Thus the foregoing example may be done as follows: 1, 1.08 1.1664, 1.259712 3 5100 £.629.8560100 629 17 1 Note. It will be sufficient to keep 5 or 6 decimal places complete, how many terms soever may be required. 2. How much will 3201. amount to at 5 cent. in 10 years, at compound interest? Σ At 5 cent. the continual multiplier is 1.05=1,50 1, wherefore 1X1.05 1.05 second term, and 1.05 multiplied by 1.05 or 1 :third term. But this is done when of 1.05 is added to itself, and so continually adding to every new term of itself produces the next succeeding term as follows, viz. By which methods we may construct Tables of the amount of 11. at 5 and 6, (or any other rate) cent. which being done, the amount of any sum for any time at compound interest is fouud by multiplying the tabular number by the principal given. QUESTIONS. 1. What sum will 4501. amount to in 3 years at 5 cent. annum ? Answ. 5201. 18s. 74d. 2. What will 2561. 10s. amount to in 7 years, at 6. cent annum? compound interest. Answ. 3851. 13s. 7 3. What will 1361. 15s. 6d. be augmented to, being forborne 20 years, at 6 cent annum? Answ. 4331. 13s. 14d. 4. What sum will 5001. amount to in 4 years, at 4 cent. annum. compound interest? Answ. 5961. 5s. 21d. SECT. II. Of Annuities or Pensions in arrear, computed at Compound Interest. To find the amount of an annuity or pension in arrear compound interest. Rule. Find the amount of the given yearly sum at compound interest, for the given years less. 1, which will be the last term of a geometrical progression, of which the given sum is the first. Then find the sum of that progression, and it is the amount of the annuity required. Otherwise. |