2. What is the cube root of 389017? Ans. 73. Ans. 319. Ans. 638. Ans. 364. Ans. 3002. Q. What is required when we are to extract the cube root of a number? How do you extract the cube root of a whole number? CASE II. § 186. To extract the cube root of a decimal fraction. RULE. Annex ciphers to the decimal, if necessary, so that it shall consist of 3, 6, 9, &c., places. Then put the first point over the place of thousandths, the second over the place of millionths, and so on over every third place to the right; after which extract the root as in whole numbers. NOTE 1. There will be as many decimal places in the root as there are periods in the given number. NOTE 2. The same rule applies when the given number is composed of a whole number and a decimal. NOTE 3. If in extracting the root of a number there is a remainder, after all the periods have been brought down, periods of ciphers may be annexed by considering them as decimals. EXAMPLES. 1. What is the cube root of ,157464? Ans. ,54. Ans. Ans. 2,35. Ans. Ans. 5. What is the cube root of ,353393243. 6. What is the cube root of 3,408862625. Ans. 1,505. 7. What is the cube root of 27,708101576. Ans. 3,026. Q. How do you extract the cube root of a decimal fraction? How many decimal places will there be in the root? Will the same rule apply when there is a whole number and a decimal? In extracting the root if there is a remainder, what may be done? CASE III. § 187. To extract the cube root of a vulgar fraction. RULE. I. Reduce compound fractions to simple ones, mixed numbers to improper fractions, and then reduce the fraction to its lowest terms. II. Then extract the cube root of the numerator and denominator separately, if they have exact roots; but if either of them has not an exact root, reduce the fraction to a decimal, and extract the root as in the last Case. Q. How do you extract the cube root of a vulgar fraction? ARITHMETICAL PROGRESSION. § 188. If we take any number, as 2, we can, by the continued addition of any other number, as 3, form a series of numbers: thus, 2, 5, 8, 11, 14, 17, 20, 23, &c., in which each number is formed by the addition of 3 to the preceding number. This series of numbers may also be formed by subtracting 3 continually from the larger number; thus, 23, 20, 17, 14, 11, 8, 5, 2. A series of numbers formed in either way is called an Arithmetical Series, or an Arithmetical Progression; and the number which is added or subtracted is called the common difference. When the series is formed by the continued addition of the common difference, it is called an ascending series; and when it is formed by the subtraction of the common difference, it is called a descending series; thus, 2, 5, 8, 11, 14, 17, 20, 23, is an ascending series. 23, 20, 17, 14, 11, 8, 5, 2, is a descending series. The several numbers are called terms of the progression the first and last terms are called the extremes, and the intermediate terms are called the means. What is the comWhat a descending What are the first Q. How do you form an Arithmetical Series? mon difference? What is an ascending series? series? What are the several numbers called? and last terms called? What are the intermediate terms called? § 189. In every arithmetical progression there are five things which are considered, any three of which being given or known, the remaining two can be determined. They are, 1st, the first term; 2nd, the last term; 3rd, the common difference; 4th, the number of terms; 5th, the sum of all the terms. Q. In every Arithmetical Progression how many things are con sidered? What are they? § 190. By considering the manner in which the ascending progression is formed, we see that the 2nd term is obtained by adding the common difference to the first term; the 3rd, by adding the common difference to th 2nd; the 4th, by adding the common difference to th 3rd, and so on; the number of additions being 1 less than the number of the term found. But instead of making the additions, we may multiply the common difference by the number of additions, that is by 1 less than the number of terms, and add the first term to the product. Hence, we have CASE I. Having given the first term, the common difference, and the number of terms, to find the last term. RULE. Multiply the common difference by 1 less than the number of terms, and to the product add the first term. Q. How do you find the last term when the first term and common difference are known? EXAMPLES. 1. The first term is 3, the common difference 2, and the number of terms 19: what is the last term? 2. A man bought 50 yards of cloth for which he was to pay 6 cents for the first yard, 9 cents for the 2nd, 12 cents for the 3d, and so on increasing by the common difference 3: how much did he pay for the last yard? Ans. $1,53. 3. A man puts out $100 at simple interest, at 7 per cent; at the end of the first year it will have increased to $107, at the end of the 2nd year to $114, and so on, increasing $7 each year: what will be the amount at the end of 16 years? Ans. $205. 4. Twelve persons agree to contribute to a charitable object in the following proportions: the first person is to give $2, the 2nd $4, the 3rd $6, and so on, each giving $2 more than the one previous: what does the last one give? Ans. $24. 5. The first term is 5, the common difference 12, and the numbers of terms 15: what is the last term? Ans. 173. § 191. Since the last term of an arithmetical progression is equal to the first term added to the product of the common difference by 1 less than the number of terms, it follows, that the difference of the extremes will be equal to this product, and that the common difference will be equal to this product divided by 1 less than the number of terms. Hence, we have CASE II. Having given the two extremes and the number of terms of an arithmetical progression, to find the common difference. RULE. Subtract the less extreme from the greater and divide the remainder by 1 less than the number of terms, the quotient will be the common difference. Q. How do you find the common difference, when you know the two extremes and number of terms? EXAMPLES. 1. The extremes are 4 and 104, and the number of terms 26: what is the common difference? We subtract the less extreme from the greater and divide the difference by one less than the number of terms. OPERATION. 26-1=25)100(4 100 Ans. 4. 2. A man has 8 sons, the youngest is 4 years old and the eldest 32, their ages increase in arithmetical progression: what is the common difference of their ages? 32-4-28: then 3-1-7)28(4. Ans. 4. 3. A man is to travel from New York to a certain place in 12 days; to go 3 miles the first day, increasing every day by the same number of miles; so that the last day's journey may be 58 miles: required the daily increase. Ans. 5 miles. |