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.-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. 72÷n 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 (19-5X4)=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 And ed, e +2d, 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+25+28+210 +213 +21=15. 5, 8, 10, 13, 15, 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=8 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 X 2 = 54 X 6 18 X 18. When a geometrical progression has but 3 terms, either of the extremes is called a third proportional to the other two; and the middle term, consequently, is a mean proportional between them. Thus, in the progression 48, 12, 3, 3 is a third proportional to 48 and 12, because 48 divided by the ratio 12, and 12 divided by the ratio = 3; or 3 X ratio 12, and 12 X ratio = 48: 12 is the mean proportional, because 12 X 12 = 48 X 3. Of the 5 properties of a geometrical progression, viz., the greater extreme, the less extreme, the number of terms, the ratio, and the sum of the terms, any three being given, the other two may be found. Let s represent the sum of the terms. 66 E 66 66 n 66 the greater extreme. the ratio. the number of terms. n when affixed as an index or exponent, represent that the term, number, or quantity, to which it is affixed, is to be raised to a power equal to the number of terms in the respective progression, &c. Any three of the five parts of a geometrical progression being given, to find the remaining two parts. EXAMPLE.The greater extreme of a geometrical progression is 162, the less extreme is 2, and there are 5 terms in the progression; required the sum of the series. +1=r; n, therefore, is equal to the number of times EXAMPLE. A farmer proposed to a drover that he would sell him 12 sheep and allow him to select them from his flock, provided the drover would pay 1 cent for the first selected, 3 cents for the second, 9 cents for the third, and so on; what sum of money would 12 sheep amount to, at that rate? NOTE. - Ratio1, 12 12 cubed =ratio ; ratio, squared = ratio &c. When it is required to find a high power of a ratio, it is convenient to proceed as follows, viz.: write down a few of the lower or leading powers of the ratio, successively as they arise, in a line, one after another, and place their respective indices over them; then will the product of such of those powers as stand under such indices whose sum is equal to the index of the required power, equal the power required." EXAMPLE.-Required the 11th power of 3. 1 2 3 4 5 3 9 27 81 243 Here 5+4+2 = 11, consequently, 243 X 81 X 9 = 11th power of 3, or To find any assigned number of geometrical means, between two given numbers or extremes. RULE. Divide the greater given number by the less, and from the quotient extract that root whose index is 1 more than the number of means required; that is, if 1 mean be required, extract the square root; if two, the cube root, &c., and the root will be the common ratio of all the terms; which, multiplied by the less given extreme, will give the least mean; and that, multiplied by the said root, will give the next greater mean, and so on, for all the means required. Or the greater extreme may be divided by the common* ratio, for the greatest mean; that by the same ratio, for the next less, and so on. EXAMPLE.Required to find 5 geometrical means between the numbers 3 and 2187. 21873 = 729, and 729 = 3, then 3X3=9X3=27 × 3 = 81 × 3 = 243 × 3 = 729, that is, the numbers 9, 27, 81, 243, 729 are the 5 geometrical means between 3 and 2187. NOTE.-The geometrical mean between any two given numbers is equal to the square root of the product of those numbers. Thus the geometrical mean between 5 and 20, = (5X20)=10. |