CASE II. When the first term of the series and the ratio are different, that is, when the first term is either greater or less than the ratio.* 1. Write down a few of the leading terms of the series, and begin the indices with a cypher: Thus, 0, 1, 2, 3, &c. 2. Add together the most convenient indices to make an index less by 1 than the number expressing the place of the term sought. 5. Multiply the terms of the geometrical series together belonging to those indices, and make the product a dividend. 4. Raise the first term to a power whose index is one less than the number of the terms multiplied, and make the result a divisor. 5. Divide, and the quotient is the term sought. EXAMPLES. 4. If the first of a geometrical series be 4, and the ratis S, what is the 7th term ? 0, 1, 2, S. Indices. 4, 12, 36, 108, leading terms. 3+2+1=6, the index of the 7th term. 108X36X12=46656 Here the number of terms multiplied are three; therefore the first term raised to a power less than three, is the 2d power or square of 4-16 the divisor. *When the first term of the series and the ratio are different, the indices must begin with a cypher, and the sum of the indices made choice of must be one less than the number of terms given in the question: because 1 in the indices stands over the second term, and 2 in the indices over the third term, &c. and in this case, the product of any two terms, divided by the first, is equal to that term beyond the first, signified by the sum of their indices. Thus, (0, 1, 2, 3, 4, &c. Indices. 21, 3, 9, 27, 81, &c. Geometrical series. Here 4+37 the index of the 8th term. 81×272187 the 8th term, or the 7th beyond the 1st. 5. A Goldsmith sold 1 lb. of gold, at 2 cents for the first ounce, & cents for the second, 32 cents for the third, &c. in a quadruple proportion geometrically; what did the whole come to? Ans. $111848, 10cts. 6. What debt can be discharged in a year, by paying 1 farthing the first month, 10 farthings, (or 2jd.) the second, and so on, each month in a tenfold proportion? Ans. 115740740 14s. 9d. Sqrs. 7. A thresher worked 20 days for a farmer, and received for the first day's work four 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 days' labor came to, supposing a pint of barley to contain 7680 corns, and the whole quantity to be sold at 2s. 6d. per bushel? Ans. £1773 7s. 6d. rejecting remainders. 8. A man bought a horse, and by agreement was to give a farthing for the first nail, two for the second, four for the third, &c. There were four shoes, and eight nails in each shoe; what did the horse come to at that rate? Ans. £4473924 5s. 34d. 9. Suppose a certain body, put in motion, should move the length of one barley-corn the first second of time, one inch the second, and three inches the third second of time, and so continue to increase its motion in triple proportion geometrical; how many yards would the said body move in the term of half a minute? Ars. 953199685623 yds. 1ft. lin. 1b.c, which is no less than five hundred and forty-one millions of miles. POSITION. POSITION is a rule which, by false or supposed num bers, taken at pleasure, discovers the true ones required. It is divided into two parts, Single or Double. SINGLE POSITION, Is when one number is required, the properties of which are given in the question. RULE. 1. Take any number and perform the same operation with it, as is described to be performed in the question. 2. Then say; as the result of the operation is to the given sum in the question:: so is the supposed number : to the true one required. The method of proof is by substituting the answer in the question. EXAMPLES. 1. A schoolmaster being asked how many scholars he had, said, If I had as many more as I now have, half as many, one-third and one-fourth as many, I should then have 148; How many scholars had he? Suppose he had 12 As 37 148 :: 12: 48 Ans. as many - 6 48 24 16 12 Proof, 148 and Result, S7 Ans. 60. 2. What number is that which being increased by 1,4, of itself, the sum will be 125 ? 3. Divide 93 dollars between A, B and C, so that B's share may be half as much as A's, and C's share three tumes as much as B's. Ans. A's share $31, B's $151, and C's $46}. 4. A, B and C, joined their stock and gained 360 dols. of which A took up a certain sum, B took 3 times as much as A, and C took up as much as A and B both; what share of the gain had each? Ans. A $40, B $140, and C $180. 5. Delivered to a banker a certain sum of money, to receive interest for the same at 6l. per cent. per annum, simple interest, and at the end of twelve years received 7311. principal and interest together; what was the sum delivered him at first? Ans. £425. 6. A vessel has 3 cocks, A, B and C; A can fill it in 1 hour, B in 2 hours and C in 4 hours; in what time will they all fill it together? Ans. 34min. 17 sec. DOUBLE POSITION, TEACHES to resolve questions by making two supp sitions of false numbers.* RULE. 1. Take any two convenient numbers, and proceed with each according to the conditions of the question. 2. Find how much the results are different from the results in the question. 5. Multiply the first position by the last error, and the last position by the first error. 4. If the errors are alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer. 5. If the errors are unlike, divide the sum of the pro ducts by the sum of the errors, and the quotient will be the answer. NOTE. The errors are said to be alike when they are both too great, or both too small; and unlike, when one is too great, and the other too small. EXAMPLES. 4 1. A purse of 100 dollars is to be divided among men, A, B, C and D, so that B may have 4 dollars more than A, and C 8 dollars more than B, and D twice as many as C; what is each one's share of the money? 1st. Suppose A 6 B 10, 2d. Suppose A 8 B 12 C 20 D 40 80 100 2d. error 20 *Those questions, in which the results are not proportional to their positions, belong to this rule; such as those in which the number sought is increased or diminished by some given number, which is no known part of the number required. 1 The errors being alike, are both too small, therefore, 2. A, B, and C, built a house which cost 500 dollars, of which A paid a certain sum; B paid 10 dollars more than A, and C paid as much as A and B both ; how much did each man pay ? Ans. A paid $120, B $130, and C $250. 3. A man bequeathed 100l. to three of his friends, after this manner: the first must have a certain portion, the second must have twice as much as the first, wanting 81. and the third must have three times as much as the first, wanting 151.; I demand how much each man must have ? Ans. The first £20 10s. second £33, third £46 10s. 4. A laborer was hired 60 days upon this condition; that for every day he wrought he should receive 4s. and for every day he was idle should forfeit 2s.; at the expiration of the time he received 71. 10s.; how many days did he work, and how many was he idle? Ans. He wrought 45 days, and was idle 15 days. 5. What number is that which being increased by its ,its, and 18 more, will be doubled ? Ans. 72. 6. A man gave to his three sons all his estate in money, viz.. to F half, wanting 50l. to G one-third, and to H the rest, which was 10l. less than the share of G; I demand the sum given, and each man's part? Ans. the sum given was £360, whereof F had £130, G £120, and H £110. |