GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. IRST, if the index of the power be even, extract the square-root of the given number; whereby it will be depressed to a power half as high; or if the index will divide by 3 without remainder, take the cube-root for a power as high; thus proceed till the required root be obtained, or an odd power result, the index of which will not divide evenly by 3. II. The root of such an odd power may be extracted thus: First, Beginning at units, point the given number into periods of as many figures each as are expressed by its index. Secondly, Find such a figure or figures, by the table of powers or by trial, as will be nearest the first of the root, whether greater or less. Thirdly, Involve the part of the root so found to the power, and take the difference between this power and as many periods of the given number as there are figures obtained of the root, and multiply this difference by the said figures for a dividend. Fourthly, Multiply the sum of the same periods and power by the integral half of the index (i. e. for a5th power, by 2, a 7th by 3, &c.) and to the product add the said power for a divisor. Fifthly, Apply the quotient, as a correction to the part of the root before found, by addition or subtraction, accordingly as that part is less or more than just. Sixthly, Repeat the operation, if greater accuracy, or more figures in the root be desired; using the root so corrected instead of the figure or figures first found, &c. EXAMPLES. 2 What is the cube-root of ? 3 What is the fourth root of 97,41 ? answer,7937005 3,1415999 5,2540S7 4 What is the sixth root of 21055,8 P/ 5 What is the seventh root of 34487717467307513182 492153794673 P answer 32017 6 What is the eighth root of 11210162813204762362464 97942460481 ? answer 13527 7. What is the ninth root of 9763796029890739602796 30298890 ? 8 What is the 365th root of 1,05 ? answer 2148,7201 1,0001336 ARITHMETICAL PROGRESSION. RITHMETICAL Progression is a rank, or series of numbers, which increase or decrease by a common difference, in which five particulars are to be observed, viz. First, The first term; Secondly, The common excess, or difference; Fourthly, The number of terms; Fifthly, The sum of all the terms. Note. In any series of numbers in arithmetical progression the sum of the two extremes will be equal to the sum of any two terms equally distant therefrom: as, 2, 4, 6, 8, 10, 12; where 2+12=14; so 4+10=14; and 6+8=14; or 3, 6, 9, 12, 15; where 3+15-18; also 6+12=18; and 9-+9=18. CASE 1. The first term, common difference, and number of terms given, to find the last term, and sum of all the terms; RULE. First, Multiply the number of terms, less 1, by the common difference, and to that product add the first term, the sum is the last term. Secondly, Multiply the sum of the two extremes by the number of terms, and half the product will be the sum of the series. EXAMPLES. EXAMPLES. 1 Bought 19 yards of shalloon, at 1d. for the first yard, 3d. for the second, 5d. for the third, &c. increasing 2d. every yard; what did they amount to ? 19-1-18 1+37-38 2 Sixteen persons bestowed charity to a poor man; the first gave 5d. the second 9d. and so on in arithmetical progression; what did the last person give; and what sum did the indigent person receive? answer the last gave 5s 5d. sum received 21 6s 8d. 3 A merchant sold 100 yards of cloth; for the first yard he received 1s. for the second 2s. for the third 3s. &c. what sum did he receive? answer 252l 10s. 4 Admit 100 stones were laid two yards distant from each other in a right line, and a basket placed two yards from the first stone; what distance must a person travel, to gather them singly into the basket? answer 11M. Sfur. 180yds. 5 Sold 54 yards of cloth; the price of the first yard was 2s. of the second 5s. &c. what was the price of the last yard, and sum of all? S the last yd. 8l 1s. answer whole sum 220ľ`1s. 6 H covenanted with K to serve him 14 years, and to have 51. the first year, and his wages to increase annually 21. during the term, what had he the last year, what on an average yearly, and what for the whole time? 311. the last year. 184. annually. 2521. whole time. CASE CASE 2. When the two extremes and number of terms are given, and the common difference of all the terms required; RULE. Divide the difference of the extremes by the number of terms, less one, the quotient will be the common difference. EXAMPLES. 1 Admit a debt be discharged at 16 several payments in arithmetical progression; the first to be 141. the last 1007. what is the common difference, and what each payment, and the whole debt? 14+100×8-9121. the whole debt. 2 A man had 10 sons, whose several ages differed alike; the youngest was 3 years old, and the eldest 48; what was the common difference of their ages ? answer 5 years. 3 There are 21 persons, whose ages are equally distant from each other; the youngest 20 years old, and the eldest 60; what is the common difference of their ages, and the age of each person? answer common difference 2 years. 20 the age of the first person. 20+2=22 of the second. of the third, &c. 22+2=24 4 A footman is to travel from Philadelphia to a certain place in 19 days, and to go but six miles the first day, increasing every day by an equal excess, so that the last day's journey may be 60 miles; what is the common difference, and distance of the journey? answer Common difference, 37 GEOMETRICAL PROGRESSION. GEOMETRICAL Progression is a series of numbers, increasing by a common multiplier, or decreasing by a common divisor, called the ratio; as, 2, 4, 8, 16, 32, &c. increase by the multiplier, 2; and 32, 16, 8, 4, 2, decrease continually by the divisor 2, &c. The last term and sum of the series are found by this RULE. Raise the ratio to the power whose index is one less than the number of terms given, which multiply by the first term, that product is the last term or greater extreme. Multiply the last term by the ratio, from the product subtract the first term, and divide the remainder by the ratio less one; the quotient will be the sum of the series. EXAMPLES. 1 Sold 24 yards of Holland, at 2d. for the first yard, 4d. the second, Ed. the third, &c. in a duplicate proportion; how much do they amount to? |