EXAMPLES. 1. The extremes are 2 and 53, and the common difference 3; what is the number of terms ? 53 2 3)51 17 I 18 Or, 532+1=18 the answer. 3 2. If That is, if 1, 3, 5, 7, 9, &c. be the numbers, Then will 1, 22, 3, 4, 52, &c. be the sums of 1, 2, 3, &c, of those terms.. For, 0+1 or the sum of I term = 1* or I I 1+3 or the sum of 2 terms = 22 or 4 Whence it is plain, that, let n be any number whatsoever, the sum of n terms will be n2. The following table contains a summary of the whole doctrine of arithmetical progression. CASES OF ARITHMETICAL PROGRESSION. : 2. If the extremes be 3 and 19, and the common differ ence 2, what is the number of terms ? Ans. 9. 3. A Solution. + 1. lta x l-atd. 2 2a-d8ds - 2a-d 2d 2 za-d8ds-d. 2 0-1 × 0+1 2s-l+a 3. A man, going a journey, travelled the first day 5 miles, the last day 35 miles, and increased his journey every day by 3 miles; how many days did he travel ? Ans. IĮ days. GEOMETRICAL Solution. a 1-n-1 x d S a 1 S 12 dxn-1 a d = common difference. 1 = greatest term. 1 GEOMETRICAL PROGRESSION. ANY series of numbers, the terms of which gradually increase or decrease by a constant multiplication or division, is said to be in Geometrical Progression. Thus, 4, 8, 16, 32, 64, &c. and 81, 27, 9, 3, 1, &c. are series in geometrical progression, the one increasing by a constant multiplication by 2, and the other decreasing by a constant division by 3. The number, by which the series is constantly increased or diminished, is called the ratio. PROBLEM I. Given the first term, the last term, and the ratio, to find the sum of the series. RULE.* Multiply the last term by the ratio, and from the product subtract the first term, and the remainder, divided by the ratio less I, will give the sum of the series. EXAMPLES. * DEMONSTRATION. Take any series whatever, as 1, 3, 9, 27, 81, 243, &c. multiply this by the ratio, and it will produce the series 3, 9, 27, 81, 243, 729, &c. Now, let the sum of the proposed series be what it will, it is plain, that the sum of the second series will be as many times the former sum, as is expressed by the ratio; subtract the first series from the second, and it will give 729-1; which is evidently as many times the sum of the first series, as is expressed by the ratio less I ; consequently 729-1 3-1 = sum of the proposed series, and is the rule; or 729 is the last term multiplied by the ratio, I is the first term, and 3-1 is the ratio less one; and the same will hold let the series be what it will. E. D. NOTE. EXAMPLES. 1. The first term of a series in geometrical progression is 1, the last term is 2187, and the ratio 3; what is the sum of the series ? NOTE 1. Since, in any geometrical series or progression, when it consists of four terms, the product of the extremes is equal to the product of the means; and when it consists of three, the product of the extremes is equal to the square of the mean; it follows, that in any geometrical series, when it consists of an even number of terms, the product of the extremes is equal to the product of any two means, equally distant from the extremes and, when the number of terms is odd, the product of the extremes is equal to the square of the mean or middle term, or to the product of any two terms equally distant from them. 1 For in each of these proportions the product of the extremes is equal to that of the means. |