(13.) Required the sum of, Too, Tooo, todos, &c. carried to 12 terms. (14.) The greater extreme of a descending series in geometrical progression is 1835008, the ratio 2, and the number of terms 19; what is the sum of the terms? Examples to Prop. 3. (15.) Required the sum of ++10‰0+70300, &c. ad infinitum. 100 TOO 27 difference between the first and second terms. square of the first term. Then, answer, Hence we may infer, that if a ball were put in motion by a force, which moved it of a league, or 1584 yards, the first minute, (or any portion of time,) of a league, or 1583 yards, the second, &c. for ever, it would go no farther than 1 mile! For, it is evident, that + &c. ad infinitum, =·3333, &c. ad infinitum; and this is equal to precisely, by the nature of vulgar fractions and infinite decimals. (16.) Required the sum of +++†, &c. ad infinitum. (17.) Required the sum of 3++27+3'1, &c. ad infinitum. (18.) If a body be put in motion by a force which moves it 10 miles the first portion of time, 9 miles in the second equal portion, and so on (in the ratio of) for ever, how many miles will it pass over? VARIATIONS. Definition. By Variations are meant, the different ways any number of things may be altered, or changed, with respect to their places. These are sometimes called Changes, Permutation, Alternation, &c. Proposition 1. To find the number of changes that can be made of any given number of things, all different from each other. Rule. Multiply continually together the numbers 1, 2, 3, 4, 5, &c., to the number of terms; and the last product will be the answer. Prop. 2. Given any number of different things, to find how many changes can be made of them, by taking any given number of them at a time. Rule. Multiply the number of things by itself less 1, and that product by the same number less 2, &c. diminishing each succeeding multiplier, by an unit, till you have made as many products (abating one) as there are things taken at a time; the last product will be the answer. Examples to Proposition 1. (1.) How many changes may be rung by 8 bells? 1x2x3x4x5×6×7×8=40320, answer. (2.) How many changes may be rung on 9 bells? (3.) An arithmetician asked a farmer with whom he lodged, what he should give him per annum for board and lodging; the farmer asked him 251. The arithmetician said that was somewhat dear; however, he would give him that sum if he would find him with board and lodging so long as he could place himself and the honest farmer's family (consisting of 6 persons) in a different position at dinner. How long might he stay for 257.? (4.) How many changes may be rung on 12 bells, and how long would they take in ringing once over, supposing 10 changes to be rung in a minute, and the year to consist of 365 days 6 hours? Examples to Prop. 2. (5.) How many changes may be rung with 4 bells out of 8? 8x8-1x8-2x8-3=8x7x6x5=1680, answer. (6.) How many changes may be rung with 7 bells out of 12? (7.) Required the number of words that can be made with 5 letters of the 26 in the alphabet, allowing any 5 letters to make a word? COMBINATIONS. Definition. By Combinations must be understood a method of taking a less number of quantities out of a greater, as often as possible, without respect to their places, and combining them together. Proposition. To find the combinations of a less number of things out of a greater, all different. Rule. Take the series 1, 2, 3, 4, 5, &c. up to the less number of things, and multiply them continually toge. ther: then take a series of as many terms, decreasing by an unit, from the greater number of things, and multiply them continually together.-Divide the latter product by the former, and the quotient will be the answer. Examples. (1.) How many combinations can be made with 5 letters out of the 26 of the alphabet? 1x2x3x4x5=120 divisor. 26×26-1X26-2×26-3 x 26-4: 7893600 dividend; and 7893600-120-65780, answer. 78 × 48 × 74×23×22 1 × 7 × 3 × 4 × 8 =13x5×2× 23 × 22=65780. (2.) A successful general was asked by his sovereign what reward he should confer upon him for his services : the general modestly asked only a farthing for every file of 10 men in a file which he could make with a body of 100 men; what sterling money will this amount to? Those who wish for further information in the doctrine of combinations, permutations, &c. may consult Mr. Emerson's Treatise on the subject, COMPOUND INTEREST BY DECIMALS. Put p= the principal or money lent. r = the ratio, or amount of £1 for a year, year, according as the payments are made yearly, t=the time, or number of payments. o the amount. Y The amounts, or values of r, in the preceding table, are calcu. Jated thus: 100: 100+3 :: 1: 1·03 =r for yearly payments. =r for yearly payments. = r for quarterly payments. This method is most commonly used.-Some writers find the value of r thus: let m = the amount of £1 for half a year, at 3 per cent., then 1.03 is undoubtedly the true amount for a year; hence, according to the principles on which the rules of compound interest are founded. =1014889, &c. =r, for 1: m :: m : 1·03. m yearly payments. = 1.03 1 : m2 :: m2 : 1·03. m 1.03 1·007417, &c. = r, for quarterly payments. 1 Or, if m the amount of £1 for-of a year, at R per cent. then r= And these values of r appear to be more correct than those given above, especially in the calculation of annuities: for which reason the following table is inserted, that the reader may use which he pleases. Mr. Ward, in his Clavis Usure, published in 1710, makes use of this method. Proposition 1. Given the principal, rate, and time, to find the amount or interest. Rule. Find the amount of £1. for the first payment, by simple interest, which involve to such a power as is denoted by the number of payments.-This power, multiped by the principal, will give the amount; from which' deduct the principal, and the remainder will be the in terest. Or, Theo. I. pxrt =a, when p, r, and t, are given. Logarithmically, log. p. + log. r × t = log. a. Prop. 2. Give the amount, rate, and time, to find the principal. Rule. As the amount of £1., at the rate and for the. time given, is to £1., so is the amount given to the principal required. Or Theo. II. == p, when a, r, and 1, are given. Logarithmically, log. a-log.rxt=log. p. Prop. 3. Given p, a, and t, to find 7. (1.) What will 2007. amount to in 6 years, at 5 per cent. per annum, compound interest, and what interest will it gain? Here the amount of £1 for the first payment is £1.05, and 1.05× |