EXAMPLEE 292 -3 S 1. The extremes are 3 and 29, and the number me terins 14, what is the cominon difference? Extremes. Number of terms less 1=15)26(2 Ans. 2. A man hau 9 sons, whose several ages differed alike, ihe youngest was 3 years old, and the oldest 35; whar was the common difference of their ayes ? Ans. 4 years 3. A man is to travel from New-London to a certain place in 9 days, and to go but s miles the first clay, increasing every day by an equal éxcess, so that the last day's journey may be 43 miles: Required the daily in. crease, and the length of the whole journey: Ans. The daily increase is 5, und ine whole jour ney 297 miles. 4. A debt is to be discharged a! 16 differenz paymena (in arithmetical progression, the first pravnucnt is esims 141, the last 100l.: What is the common dillerence, and the sum of the whole debt? Ans. 51. 14s. 8d.common difference, and 912. ihe whole PROBLEM III. Given the first term, layt term, and common difference to find the number of terms. RULE. Divide the difference of the extremes by the common difference, and the quotient increased by i is the number of terms. EXAMPI.ES. 1. It the extremes be s anal 45, aur the comron dit frence 2; what is the number of terms ? Ans. 02. 2. A man going a journey, travalled the first day live miles, the last day 45 iniles, and cach day increased his journey by 4 miles; how many days did lie travel and how firi a 1:6. 11 days, and the whole distance travelled 275 miles GEOMETRICAL PROGRESSION, Is when sny rank or series of numbers increased by one common multiplier, or decreased by one common divisor ; as 1, 2, 4, 8, 16, &c. increase by the multiplier 2; and 27, 9, 3, 1, decrease by the divisor 3. PROBLEM I. The first term, the last term (or the extremes) and the ratio given, to find the sum of the series, RULE. Multiply the last term by the ratio, and from the product subtract the first term; then divide the remainder by the ratio, less by 1, and the quotient will be the sum of all the terms. EXAMPLES. 1. Jf the scries be 2, 6, 16, 54, 162, 486, 1458, and the ratio 3, what is its sum total ? SX1458-2 =2186 the Ansicer. S-1 2. ale extremes of a geometrical series are 1 and 65536, and the ratio 4; what is the sum of the series : Ans. 87381. PROBLEM II. Given the first term, and the ratio, to find any other tern assigned.* CASE I. When the first term of the series and the rauw are equal. *As the last terin in a long series of numbers is very te. dious to be found by continual multiplications, it will be necessary for the readier finding it out, to have a series of numbers in arithmetical proportion, called indices, whose common difference is 1. † When the first term of the series and the ratio are equal, the indices must begin with the unit, and in this case, the 1 Write down a few of the leading terms of the series, and place their indices over them, beginning the indices with an unit or 1. 2. Add together such indices, whose sum shall make up the entire index to the sum required. 3. Multiply the terms of the geometrical series belong. ing to those indices together, and the product will be the term sought. EXAMPLES. 1. If the first be 2, and the ratio 2 ; what is the 13th term. 1, 2, 3, 4, 5, indices. Then 5+5+3=13 2, 4, 8, 16, 32, leading terms. 32x32x8=8192 Aus. 2. A draper sold 20 yards of superfine cloth, the first yard for Sd. the second for 9d. the third for 27d. &ic. in triple proportion geometrical ; what did the cloth come to at that rate : The 20th, or last term is 5486784401d. Then S +3486781401.-3 =5930176600d. the 'suin of all S-1 the terms (by Prob. I.) equal to £21792402 10s. Ins. 3. A rich miser thought 20 guineas a price too much for 12 fine horses, but agreed to give 4 cents for the first, 16 cents for the second, and 64 cents for the third horse, and so on in quadruple or fourfold proportion to the last what did they come to at thet rate, and how much diu they cost per head, one with another ? Ans. The 12 horses came to $223696, 20cts, and the average price was 818641, S5cts. per head. product of any tro terms is equal to that teru, signified by the sum of their indices. Thus, 5 1 2 3 4 5 kc. Indices or arithmetical series. 22 4 8 16 52 X c. geometrical series. Vou, 3+2 5. the inder of the fifth term, and 4 X8 S2 ihe jifth tern. CASE II.' rent, that is, when the first term is eigther greater or 1. Write down a few of the leading terms of the series, and begin the indices with a cypher: Thus, |, 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. 3. Multiply the terms of the geometrical series together belonging to those indices, and make the product a dividend. 4. Raise t!ie first term to a power whose index is one. ess than the number of the terms multiplied, and make the resulta divisor. 5. Divide, and the quotient is the term sought. EXAMPLES. 4. If the first of a geonietrical series be 4, and the ratio 3, what is the 7tii term ? 0, 1, 2, S, Indices. 3+2+1=6, the index of the 7th term. =2916 the 7th term required 16 Here the number of ternis multiplicd are three; thier fore the first terin raised to a power less than three, is the 22 power or square of 4=16 the divisor. *When the first terin 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 I in the indices stands over the second term, and 2 in the indices over the third tern, fc. and in this case, the product of any tio terins, divided by the first, is equal to that terin beyond the first, signified by the sum of their indices. Thus, 50, 1, 2, S,' 4, Sc. Indices. {1, 3, 9, 27, 81, &c. Geometrical series. Here 4+3=7 the inde.r of the 8th term. 8127=2187 the 8th term, or the 7tir beyond the 5. A Goldsmith sold 1 lb. of gold, at 2 cents for the first ounce, 8 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 24d.) the second, and so on, each month in a tenfold proportion ? Ans. £ 115740740 148. 9d. 3qrs. 7. A thresher worked 20 days for a farmer, and receive ed for the first day's work four barley-corns, for the second 12 barley-corns, for the third 36 barley-corns, and so on m 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 Tur 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. 33d. 9. Suppose a certain body, put in motion, should move the length of one barley-corn the first second of time, onc inch the second, and three inches the third second of time, and so continue to increase its motion in triple pro portion geometrical; how many yards woulil the said body move in the term of half a minute ? Ar s. 933199685623 yds. ift. 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 numbers, 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. |