must be always one less than the number of terms given in the question, as 1 in the indices stands over the second term. EXAMPLES. 1. A boy agrees for 16 oranges, to pay only the price of the last, reckoning a cent for the first, two cents for the second, four cents for the third, &c. doubling the price to the last : How much did he give for them ? Ans. $327.68. 2. A butcher meets with a drover who had 23 oxen; he demanded the price of them, and was answered 161. a head: the butcher bids him 151. a head, and would take all : the drover tells him it could not be taken ; but if he would give what the last ox would come to, at a farthing for the first, and doubling it to the last, he should have all : Required the price of the oxen? Ans. 43691. 18. 4d. II. In any series, not proceeding from unity, the ratio and first term being given, to find any remote term. RULE-Proceed as in the last case, only observe to divide every product by the first term, and the quotient will be the term required. EXAMPLES. 3. A sum of money is to be divided among 8 persons, the first to have $20, the second 860, and so on in triple proportion : What will the last have ? Ans. $43740. 4. A gentleman dying left nine sons, to whom and to his executors, he bequathed his estate in manner following, viz: To his executors 501. his youngest son was to have as much more as the executors, and each son to exceed the next younger by as much more : What was the eldest son's fortune ? Ans. 256002. III. When the first term, ratio, and number of terms are given, to find the sum of all the terms. RULE- Find the last term as before, from which subtract the first, divide the remainder by the ratio, less one, and to that quotient add the last term, gives the sum required. EXAMPLES. 5. A man bought a horse, and by agreement was to give a farthing for the first nail, three for the second, &c. there were four shoes, and in each shoe 8 nails: What was the amount of the horse ? Ans. 9651146816931. 13s. 4d. 6. A gentleman married his daughter on New-Year's day, and gave her husband an English guinea towards her fortune, promising to double it on the first day of every month for one. year. What was her fortune in federal money? Ans. $19110. 7. A miller buys 12 stacks of wheat, and was to pay ? cents for the first stack, 6 cents for the next stack, tripling the price for every following stack: What sum did'he pay ? Ans. $5314.40. 8. A grain of wheat being sown, produces 7 grains, which are sown again and yield the same increase : Required how much it will amount to in 12 years, if the whole crop be always sown and yield the same increase ? and how many bushels, allowing 700000 grains to a bushel ? Ans. 13841287201 grains, 19773 bushels. 9. If the posterity of Noah, which consisted of six persons at the flood, increased so as to double their number in 20 years, how many inhabitants would there be in the world two years before the death of Shem, who lived 502 years after the flood ? Ans. 201326592. 10. The Indian who invented the game of chess, is said to have asked of the King, who promised him any reward he should demand, that he might have one grain of wheat for the first point of the board, two for the second, and so on, in geometrical progression for all the 64 points. The King considering that as a small matter, ordered the wheat to be given him. How much will it amount to? 18446744073709551615 grains. Ans.} 2635249153387 bushels. IV. Of any decreasing series in Geometrical Progression, whose last term is a cipher, to find the sum of those series. Rule-Divide the square of the first term by the difference between the said first term, and the second term in the series, the quotient will be the sum of the series. EXAMPLES. 11. A great ship pursues a small one, steering the same way, at the distance of 4 leagues from it, and sails twice as fast as the small ship: Required how far the great ship must sail before it overtakes the lesser ? Ans. 24 miles. 12. Suppose a ball to be put in motion by a force which drives it 12 miles the first hour, 10 the second, and so on continually decreasing in proportion of 12 to 10 to infinity : What space would it move through ? Ans. 72 miles. PERMUTATION. Permutation is a rule for finding how many different ways any given number of things may be varied in position, thus, a b c, a cb, bac, b c a, ca b, c b a, are 6 different positions of 3 letters. RULĖ-Multiply all the terms of the natural, series contin. ually, from one to the given number inclusive, the last pro. duct will be the changes required. M м EXAMPLES 1. In how many different positions can 6 persons place themselves at table ? 1X2X3X4X5X6=720 positions. 2. Required the number of changes that may be rung on 12 bells, and the time in which they may be rung, allowing 3 seconds to every Ans. S 45 yrs. 195 days, 18 ho. 3. What time will it require for 8 boarders to seat themselves every day differently at dinner ? Ans. 110 yrs. 142 days. 4. How many variations will the 26 letters of the alphabe admit of ? Ans. 403291461126605635584000000. COMBINATION. Combination discovers how many different ways a less number of things may be combined out of a greater ; thus out of the numbers, 7, 8, 9, are 3 different combinations of 2, viz : 78, 79, 89. Rule-Take a series proceeding from, and increasing by one, up to the number to be combined ; and another series of as many places, decreasing by one, from the number out of which the combinations are to be made ; multiply the first continually for a divisor, and the latter for a dividend ; the quotient will be the answer. EXAMPLES. 1. How many combinations of 5 letters in 10? 10 X9X8X7X6 252 Answer. 1X2X3 X4 X5 2. A General who had often been successful in war, was asked by his King what reward he should confer upon him for his services; the general only desired a farthing for every file, of 10 men in a file, which he could make with a body of 100 men : Required the amount in pounds sterling ? Ans. 180315723501. 98. 2d. 3. How many different ways may a butcher choose 50 sheep out of a flock consisting of 100, so as not to make the same choice twice ? Ans. 10891306544874079257172497256. SIMPLE INTEREST BY DECIMALS. Here are five letters to be observed, viz. P, or © any principal or sum put to interest. I, or is the interest. T, or t = the time of the principal's continuance at interest A, or a = the amount, or principal and its interest. R, or r = the ratio, or rate per centum per annum. Note.-I'he ratio, is the simple interest of 1l. or $. for 1 year, at any given rate per cent and is thus found, viz. As {100 : $ $ 54 : :1:.055 the ratio at 54 per cent. per annum. 1. Given p, t, and r, to find i. -RULE. PXtXrsi. EXAMPLES. 1. What is the interest of $567 50 cents for 9 years at 6 per cent per annum ? Ans. $306 45 cents. 2. What is the interest of 1721. 10s. for 7 months at 7 per cent per annum ? Ars 71. Os. 1c_d.. 3. How much is the interest of $700 for 1 year and 73 days at 6 per cent per annum ? Ans. $50 40 cents. II. Given p, t, and r, to find -Rule. PXtXr+p=a. EXAMPLES. 4. What will $584 33 cents amount to in three years at 6 per cent per annum ? Ans. $689 51 cents. 5. What is the amount of 4401. for 7 years at 7 per cent per annum? Ans. 6551. 12s. i III. Given r, t, and i, o find p. -RULE. EXAMPLES. 6. I demand what principal, being put to interest for 9 years, will gain $306 45 cents at 6 per cent per annum ? Ans. $567 50 cents. 7. I demand what principal, being put to interest for 7 months, will gain 71. Os. 103d. at 7 per cent per annum ? Ans. 1721. 10s. IV. Given a, r, and t, to find p.-RULE. tXr+1 EXAMPLES. 8. What principal being put to interest, will amount to $689 51 cents in 3 years at 6 per cent per annum? Ans. $5841 9. What principal being put to interest, will amount to 6551. 128. in 7 years at 7 per cent per annum ? Ans. 4401. i V. Given, p, 2, and r, to find t. RULE. por EXAMPLES. 10. In what time will $567 50 cents, gain $306 45 cents, at 6 per cent per annum ? Ans. 9 years 11. In what time will 1721. 108. gain 71. Os. 104d. at 7 per cent per annum ? Ans. 7 months. 136 ANNUITIES OX PENSIONS, &c. IN ARREARS. ap VI. Given, p, i, and r, to find t. RULE. PXr EXAMPLES. 12. In what time will $700 amount to $750 40 cents at 6 per cent per annum? Ans. 1 year 73 days. 13. In what time will 1721. 108. amount to 1791., 108. 103d. at 7 per cent per annum ? Ans. 7 months. i VII. Given, p, i, and t, to find r. Rule PXt EXAMPLES. 14. At what rate per cent will $567 50 cents gain $306 45 cents in 9 years ? Ans. 6 per cent. 15. At what rate per cent will 1721. 10s. gain 71. Os. 1c1d. in 7 months ? Ans. 7 per cent. ap VIII. Given, p, a, and t, to findr. -RULE. pxt EXAMPLES. 16. At what rate per cent will $700 amount to $750 40 cents in one year and 73 days ? Ans. 6 per cent. 17. At what rate per cent will 1721. 10s. amount to 1791. 108. 10 d. in 7 months ? Ans. 7 per cent. ANNUITIES OR PENSIONS, &c. IN ARREARS. Annuities or Pensions, &c. are said to be in Arrears when they are payable or due, either yearly, half-yearly, or quarterly, and are unpaid for any number of payments. Here u, represents the Annuity, Pension, or yearly rent, a, 2 t, r, as before, and I. Given, l, r, and t, to find a. Rule. txt Xu.mt :+Xu=a. 2 EXAMPLE. 18. If 2501. yearly rent, be forborne 7 years, what will it amount to in that time at 6 per cent ? Ans. 20652. For exercise the scholar may work the same, payable in half-yearly, and quarterly payments. Note-If the pay- Shalf-yearly then use Str, tu, and 2 t. inents are made quarterly 14r, . u, and 4 t. II. Given, a, r, and t, to find u.-RULE. txtxr-xrt2t EXAMPLE. 19. If a pension amount to 20651. in 7 years, at 6 per cent. What is the pension ? Ans. 2501. Xuxt 2 a |