4. The difference of two numbers is 10; and their product is to the sum of their squares as 6: 37. the numbers? What are 5. The product of two numbers is 136; and the difference of their squares is to the square of their difference as 25: 9. What are the numbers? Ans. 8 and 17. 6. As two boys were talking of their ages, they discovered that the product of the numbers representing their ages in years was 320, and the sum of the cubes of these same numbers was to the cube of their sum as 7:27. What was the age of each? Ans. Younger, 16; elder, 20 years. 7. As two companies of soldiers were returning from the war, it was found that the number in the first multiplied by that in the second was 486, and the sum of the. squares of their numbers was to the square of the sum as 13:25. How many soldiers were there in each company? Ans. In 1st, 27; in 2d, 18. 8. The difference of two numbers is to the less as 100 is to the greater; and the same difference is to the greater as 4 is to the less. What are the numbers? NOTE. Multiply the two proportions together. (Theorem XIII.) SECTION XXII. PROGRESSION. 216. A PROGRESSION is a series in which the terms increase or decrease according to some fixed law. 217. The TERMS of a series are the several quantities, whether simple or compound, that form the series. The first and last terms are called the extremes, and the others the means. ARITHMETICAL PROGRESSION. 218. An ARITHMETICAL PROGRESSION is a series in which each term, except the first, is derived from the preceding by the addition of a constant quantity called the common difference. 219. When the common difference is positive, the series is called an ascending series, or an ascending progression; when the common difference is negative, a descending series. Thus, a, a +d, a + 2d, a + 3d, &c. is an ascending arithmetical series in which the common difference is d; and is a descending arithmetical series in which the common difference is - d. 220. In Arithmetical Progression there are five elements, any three of which being given, the other two can be found: 1. The first term. 2. The last term. 3. The common difference. 4. The number of terms. 5. The sum of all the terms. 221. Twenty cases may arise in Arithmetical Progression. In discussing this subject we shall let 222. The first term, common difference, and number of terms given, to find the last term. In this Case a, d, and n are given, and 7 is required. The successive terms of the series are a, a + d, a + 2 å, a + 3d, a + 4d, &c.; that is, the coefficient of d in each term is one less than the number of that term, counting from the left; therefore the last or nth term in the series is in which the series is ascending or descending according as d is positive or negative. Hence, RULE. To the first term add the product formed by multiplying the common difference by the number of terms less one. 1. Given a = 4, d 2, and n = 9, to find l. l=a+ (n-1) d = 4+ (9 − 1) 2=20, Ans. 2. Given a = 7, d= 3, and n = 19, to find l. Ans. 61. 3. Given a = 29, d —— 2, and n = 14, to find l. Ans. I = 3. 4. Given a = 40, d= 10, and n = 100, to find l. 223. The extremes and the number of terms given, to find the sum of the series. In this Case a, l, and n are given, and S is required. Now S = a + (a + d) + (a + 2d) + (a + 3d) + +1 S = 1 + ( l − d) + ( l − 2 d) + ( l − 3 d) + . + a + (a + 1) or, inverting the series, Adding these together, 2 S = (a + i) + (a + 1) + (a + 1) + (a + 1) + And since (a+1) is to be taken as many times as there are terms, 2 Sn (a+1) hence or S=(a+1). Hence, RULE. Find one half the product of the sum of the extremes and the number of terms. -- NOTE. If in place of the last term the common difference is given, the last term must first be found by the Rule in Case I. 1. Given a = 3, 7 = 141, and n = 26, to find S. n 26 S = '" (a + 1) = 2/25 (3 + 141) = 1872, Ans. 2. Given a = ,725, and n = : 63, to find S. 7. Given a =— 5. Given a = }, d , and n = 3, to find S. 11, to find S. 25, to find S. CASE III. 224. The extremes and number of terms given, to find the common difference. In this case a, l, and n are given, and d is required. Divide the last term minus the first term by the number of terms less one, and the quotient will be the common difference. 1. Given a = 5, 747, and n = 7, to find d. NOTE. This rule enables us to insert any number of arithmetical means between two given quantities; for the number of terms is two greater than the number of means. Hence, if m = ber of means, m + 2 =n, or m +1 = n - 1, and d the num = m+1 Having found the common difference, the means are found by adding the common difference once, twice, &c., to the first term. |