EXAMPLE.-Required the fifth root of 432040.0354. 432040.03540 ( 13.4+. Ans. 15=1 14 X 5) 33 135 371293 = For instructions touching special cases, see NOTES relative to the extraction of the square root, and to the extraction of the cube root. A series of three or more numbers, increasing or decreasing by equal differences, is called an arithmetical progression. If the numbers progressively increase, the series is called an ascending arithmetical progression; and if they progressively decrease, the series is called a descending arithmetical progression. The numbers forming the series are called the terms of the progression, of which the first and the last are called the extremes, and the others the means. The difference between the consecutive terms, or that quantity by which the numbers respectively increase upon each other, or decrease from each other, is called the common difference. Thus, 3, 5, 7, 9, 11, &c., is an ascending arithmetical progression, and 11, 9, 7, 5, 3, is a descending arithmetical progression. In these progressions, in both instances, 11 and 3 are the extremes, of which 11 is the greater extreme, and 3 is the less extreme. The numbers between these, (9, 7, 5,) are the means. In every arithmetical progression, the sum of the extremes is equal to the sum of any two means that are equally distant from the extremes; and is, therefore, equal to twice the middle term, when the series consists of an odd number of terms. Thus, in the foregoing series, 3+11=5+ 9 = 7 X 2. The greater extreme, the less extreme, the number of terms, the common difference, and the sum of the terms, are called the five properties of an arithmetical progression, of which, any three being given, the other two may be found. Let s represent the sum of the terms. 66 E 66 the greater extreme. the common difference. The extremes of an arithmetical progression and the number of terms being given, to find the sum of the terms. EXAMPLE. What is the sum of all the even numbers from 2 to 100, inclusive? 102 X 50 22550. Ans. EXAMPLE. — How many times does the hammer of a commcn clock strike in 12 hours? (2e+n - 1 × d) × 1 n = sum of the terms. The greater extreme, the common difference, and the number of terms of an arithmetical progression being given, to find the less extreme. E-(dXn-1) = less extreme. EXAMPLE. A man travelled 18 days, and every day 3 miles farther than on the preceding; on the last day he travelled 56 miles; how many miles did he travel the first day? √ (E × 2 + d) 2 — sx dx8+d= less extreme, when 2 √(2E+d) — 8 sd is equal to, or greater than d. √ (2 E + d)2 — 8 s d d = less extreme, when 2 - The extremes of an arithmetical progression and the common difference being given, to find the number of terms. E-ed+1: = number of terms. EXAMPLE. As a heavy body, falling freely through space, descends 16 feet in the first second of its descent, 48 feet in the next second, 80 in the third second, and so on; how many seconds had that body been falling, that descended 305 feet in the last second of its descent? 30512-16=2891 ÷ 327=9+1=10 seconds. Ans. √ (2 er d)2 + 8 s d+d—2e = number of terms. 2 2s÷E+(2 E + d) s −8 s d+d: 2 == number of terms when √ (2 E + d)2 — 8 s d is equal to, or greater than d. 2 s÷E+ (2 E + d)2 — 8 s d d = number of terms when 2 √(2E+d)-8 sd is less than d. SX 2 E+c = number of terms. The extremes of an arithmetical progression, and the number of terms being given, to find the common difference. EXAMPLE. One of the extremes of an arithmetical progression is 28 and the other is 100, and there are 19 terms in the series; required the common difference. EXAMPLE. common difference. The less extreme of an arithmetical progression is 28, the sum of the terms 1216, and the number of terms 19; required the 7th term in the series, descending. 1216 X 219 = 128 = = sum of the extremes. 128-28=100= greater extreme. 100 28-72= difference of extremes. 72÷n- −1 (18)=4= common difference. 100 (7—1×4)=76= 7th term descending. Ans. Required the 5th term from the less extreme, in an arithmetical progression, whose greatest extreme is 100, common difference 4, and number of terms 19. 100 (19 — 5 × 4) = 44. Ans. To find any assigned number of arithmetical means, between two given numbers or extremes. RULE. -Subtract the less extreme from the greater, divide the remainder by 1 more than the number of means required, and the quotient will be the common difference between the extremes; which, added to the less extreme, gives the least mean, and, added to that, gives the next greater, and so on. Or, E e÷m+1=d, E being the greater extreme, e the less extreme, m the number of means required, and d the common differ ence. And ed, e2d, e+ 3 d, &c.; or, E-d, E-2d, E-3 d, &c., will give the means required. EXAMPLE. - Required to find 5 arithmetical means between the numbers 18 and 3. 18-3-15÷6=21, and 3+21=5+25=8+25=103 +21=13+21=151⁄2. 5, 8, 101, 13, 152, therefore, are 5 arithmetical means, between the extremes, 3 and 18. NOTE. The arithmetical mean between any two numbers may be found by dividing the sum of those numbers by 2; thus, the arithmetical mean of 9 and 8 is (9+8)÷2=81. GEOMETRICAL PROGRESSION. A series of three or more numbers, increasing by a common multiplier, or decreasing by a common divisor, is called a geometrical progression. If the greater numbers of the progression are to the right, the progression is called an ascending geometrical progression, but, on the contrary, if they are to the left, it is called a descending geometrical progression. The number by which the progression is formed, that is, the common multiplier, or divisor, is called the ratio. The numbers forming the series are called the terms of the progression, of which the first and the last are called the extremes, and the others the means. The greater of the extremes is called the greater extreme, and the less the less extreme. Thus, 3, 6, 12, 24, 48, is an ascending geometrical progression, because 48 is as many times greater than 24, as 24 is greater than 12, &c.; and 250, 50, 10, 2, is a descending geometrical progression, because 2 is as many times less than 10, as 10 is less than 50, &c. In the first mentioned series, (3, 6, 12, 24, 48,) 48 is the greater extreme, and 3 is the less extreme; the numbers 6, 12, 24 are the means in that progression. So, too, of the progression 250, 50, 10, 2; 250 and 2 are the extremes, and 50 and 10 are the means. In the first mentioned progression, 2 is the ratio, and in the last, or in the progression 2, 10, 50, 250, 5 is the ratio. In a geometrical progression, the product of the two extremes is equal to the product of any two means that are equally distant from the extremes, and, also, equal to the square of the middle term, when the progression consists of an odd number of terms. Thus, in the progression 2, 6, 18, 54, 162; 162 × 2 = 54 × 6 = 18 X 18. = When a geometrical progression has but 3 terms, either of the |