17. Given the sum s, of three numbers, of which the square of the greatest is equal to the squares of the other two, and also the continued product p, of the three numbers; to find the numbers. 18. Let P, 48 be the given product of the two lesser numbers, the rest as before; to find the numbers. Ans. The greatest is s2 — 2p, and the sum of the two lesser ones is 82+2p 28 " 28 19. Let, as before, the square of the greatest be equal to the squares of the other two, and the square of the middle one equal to the product of the greatest and least, and let the sum s, of the three be given; to find each of them. 20. Suppose still the square of the greatest equal to the squares of the other two, and let the difference of the squares of the two least be equal to the product of the greatest by a given multiplier m, also the difference of the two least is given =d; to find the numbers. A SERIES of quantities, which increase or decrease by a common difference, is called an Arithmetical Progression; as, 2, 5, 8, 11, &c., or 88, 85, 82, &c. A series of quantities, which increase by a constant multiplier, or decrease by a common divisor, is called a Geometrical Progression; as, 2, 8, 32, 128, &c., or 567, 189, 63, &c. The greatest and least terms are called the Extremes, and the other terms the Means. ARITHMETICAL PROGRESSION. If a represent the least term, y the greatest, d the common. difference, and n the number of terms, any arithmetical progression may be expressed thus: a, a+d, a +2d, a+3d, &c. ascending; or y, yd, y-2d, y-3d, &c. descending. From these expressions it appears that the coefficient of d in any term is less by 1 than the number of that term. PROP. I.-The difference between the extremes is equal to the common difference, multiplied by the number of terms wanting one. For the coefficient of d in the nth term is n—1. Cor. Hence y=a+(n−1)d, and a =y· - (n − 1)d. PROP. II.—The sum of the extremes is equal to the sum of any two terms equally distant from them. For any term exceeds the least, as much as its corresponding term is less than the greatest. Thus, if half the series ascend from a, while the other half descends from y, the whole will be a, a+d, a +2d, &c., y—2d, y―d, y; where the sum of any two corresponding terms is a+y. Cor.-The double of any term is equal to the sum of any two terms equally distant from it. PROP. III.—The sum of any number of terms in arithmetical progression is equal to the sum of the extremes multiplied by half the number of terms. For by adding the extremes, and every two equally distant from them, we obtain equal sums, of which the number is half the number of terms of the series. n Cor. 1.-Hence if s = sum of the series, s=(a+y) • Cor. 2.-If the number of terms be odd, and m the middle one, then snm; for 2m=a+y. Cor. 3.-In a series of natural numbers, 1, 2, 3, &c. n, the sum snx n+1 2 ; for n is the greatest term, and n+1 the sum of the extremes. Cor. 4.-In a series of even numbers, 2, 4, 6, &c., s= n(n+1); for this series is 2 × (1+2+3), &c. as Cor. 5.-In a series of odd numbers, beginning at 1, 1, 3, 5, &c., s=n2; for the sum of the extremes is double the number of terms. 1. Required the 12th term of the series 5, 8, 11, &c. Here n= = 12, a = 5, d=3; therefore y=5+11x3=38. 2. Required the 7th term of the series 182, 178, 174, &c. Here n=7, y=182, d=4; therefore a = 182 — 6 × 4 =158. 3. Required the sum of 12 terms of the series 3, 8, 13, &c. Here a=3, d=5, n=12, y=3+11 × 5 = 58, and S=(58+3)6=366. 4. Required the sum of 14 terms of the series 89, 85, 81, &c. Here a 89-13 × 437, and s= (89+37)7882. From these propositions any two of the five things mentioned may be found, if the other three be given. The theorems for finding them are expressed in the following Table. USE OF THe table. 1. Let the least term be 7, the common difference 2, and the sum of the series 567. Required the greatest, and the number of terms. √(567 × 8 ×2+14—2|2) = √(9072+144)=√9216=96, 96-2 47, the greatest term; and and 96-14+2 =21, the 2. Given the least term 5, the number of terms 30, and the sum of the series 1455; to find the greatest term and the common difference. 1455 × 2 30 ·5=92 the greatest, 1455-5x30 3 the differ ence. 3. Given the common difference 4, the number of terms 20, and the sum of the series 1240; to find the least and greatest If a be the least term of a geometrical progression, y the greatest, r the common multiplier or divisor, called the common ratio, and n the number of terms, such a series, if ascending, may be expressed thus, a, ar, ar2, ar3, &c., or if descending, thus, y, y y У r is one less than the number of the term. of PROP. I. The greatest term of a geometrical progression is equal to the least term, multiplied by that power of the common ratio, of which the exponent is the number of terms wanting one. For in the nth term, the exponent of r is n — - 1. Required the 8th term of the series 2, 6, 18, &c. Here a=2, r = 3, n = 8; therefore 2 × 374374. PROP. II. The product of the extremes is equal to the product of any two terms equally distant from the extremes. For a xy=arxar2 x 1/1, &c. Cor. 1.-The square of any term is equal to the product of any two terms equally distant from it. Cor. 2.-If there be four terms, the product of the means, divided by either extreme, gives the other; and if there be three terms, the square of the mean, divided by either extreme, gives the other. 1. Required a third proportional to 85 and 425. Ans. 2125. a fourth proportional to 18, 54, 162. 486. 2. PROP. III.—If the sum of a geometrical progression be multiplied by the common ratio, and the series be subtracted from the product, the remainder will be equal to the excess of the product of the highest term by the ratio, above the least term. For the whole series, except the least term, will be included in the product. Thus, if a+ar+ar2, &c.+ ++ y = s 72 T y be multiplied by r, it becomes ar+ar2, &c. ++y+yr=sr; and subtracting the original series, we obtain yr-asr—s. yra a(r"—1) Whence s= = Cor. 1.-The difference between any two adjacent terms is equal to the less multiplied by the ratio, wanting one. Thus, ar3 — ar2 = ar2 × (r− 1). Wherefore, if the difference of the extremes be multiplied by the greatest term but one, and divided by the difference between the two greatest terms, the quotient will be the sum of all the terms except the 1-1 = a In this formular may represent any quantity, integral or fractional, except unity. If r = 1, there could be no progression; for every a(1—1) ахо power of 1 is 1, and therefore the formula would be very improper expression. When a is multiplied by a quantity less than 1, the product is less than the multiplicand; and the less that the multiplier is taken, the less will the product be; so that a×0=0, or less than any quantity. Again, when a is divided by a quantity less than 1, the quotient is greater than a; and the less that the divisor is taken, the greater will the quotient be: therefore will be infinitely great, or a greater than any quantity. To avoid this absurdity, divide first by the denominator, and then affix values to the quantities. If ar". -a be divided by 7-2 — 1, the quotient is ar2-1+ar2¬2+ar2-3, &c.; and if r = 1, it will be a(1+1+1+1, &c.) =na, which, though not a geometrical progression, is a determined quantity. In like manner would be if x were a; but if we divide first, the quotient will be xa x+a, which is = 2a, when x = a. And many other cases may occur like these. F |