their geometrical mean by, and the geometrical mean exceeds their harmonical mean by 5. Find the numbers. ± √AXH. 8. Prove that G = 9. When x, y, z, are in H. P. show that x 10. The sixth term of a H. P. is 1 and the tenth term §. Find the first five terms. REVIEW EXERCISE XXIV 1. Find the sum of the series, 1, 3, 5, to n terms. 2. Three numbers, whose sum is 24, are in A. P., but. if 3, 4, and 7, are added to them respectively, these sums are in G. P. Find the numbers. 3. Find the sum of eight terms of a G. P. whose last term is 1 and fifth term 1. 4. Find the last term and the sum of the series 32, -16, +8,. to seven terms. . 5. The sum of three numbers in A. P. is 27, and their product is 693. Find the numbers. 6. Insert six geometrical means between and 12. 7. There are four numbers, the first three of which are in G. P. and the last three in A. P. The sum of the first and last is 14 and the sum of the second and third is 12. Find them. 8. In a G. P. of ten terms, the fourth and seventh terms are 5 and 625 respectively. Find the remaining elements. 9. How many consecutive odd numbers, beginning with 7, must be taken to give 775? 162, find l and n. 10. Given a = = 1 s = 91 11. The arithmetical mean between two numbers is 11, and their harmonical mean is 40 4; find the numbers. 12. Insert five harmonical means between 2 and -3. 13. The difference between two numbers is 48, and their arithmetical mean exceeds their geometrical mean by 8. Find the numbers. 14. A body, rolling down an inclined plane, goes 6 feet the first second, 18 feet the next, 30 feet the third and so on. How many seconds will it take to roll 486 feet? 15. What number added to 1, 13, and 73, will give results in G. P.? n+1 n+2 n + 3 + + n n n to n terms. 17. Write three progressions of six terms, one arithmetical, one geometrical, and one harmonical, in each of which the second term is -6, and the fourth term −54. 18. Find the value of 0.11003003. . 19. In a "potato race," 50 potatoes are placed in a straight line 4 feet apart, the first being 5 feet from the basket. How far must a contestant travel in bringing them to the basket one at a time? 20. Find the value of 2.34848. . 21. Find the sum of all the numbers between 100 and 600 that are divisible by 11. 22. Insert between 6 and 16, two numbers, such that the first three of the four shall be in A. P., and the last three in G. P 23. If the fourth term of a G. P. is and the seventh 27, how many terms beginning with the first must be taken so that their sum may be 4898? 09 24. Divide 9 into three parts in G. P. such that the sum of the first two is to the sum of the last two as 3 is to 2. 25. Find a G. P. of which the sum of the first two terms is 22 and the sum to infinity 4. 26. In boring a well 400 feet deep, the cost is 27 cents for the first foot and an additional cent for each subsequent foot. What is the cost of boring the last foot, and the entire cost? 27. Find the sum of all even numbers from 2 to 50 inclusive. 28. Find the limiting sum of the series, 1 + 1 + 25 +125 to infinity. 29. Find the value of the decimal 0.922828... CHAPTER XXV SIMPLE INDETERMINATE EQUATIONS 182. A single equation involving two unknown numbers may have an unlimited number of solutions if no other conditions are imposed, for, if any value is assigned to one of the two unknown numbers, a corresponding value of the other may be found. Thus if, 2x + 3y = 6, any value assigned to x will have a corresponding value of y. Any equation that has an indefinite number of solutions is called indeterminate. In the case of simultaneous equations, if the number of equations is less than the number of unknown numbers, the system will also have an indefinite number of solutions, and hence is indeterminate. By eliminating one of the unknowns, we obtain a single equation containing two unknowns and therefore indeterminate. The values of the unknown numbers in an indeterminate equation are dependent upon each other, that is the values are corresponding values and must agree with the condition stated by the equation. If additional conditions are imposed upon the values of the unknowns, the number of solutions may be limited. For instance, let us impose the condition that the values of the unknowns must be positive integers. 1. Solve, in positive integers, 3x + 4y = 25. Divide the equation by the smaller coefficient, 3, and express the quotient in integers and fractions. Transpose the fractions to the left side of the equation and the integral forms to the right. As x and y must be positive integers, the right side of the equation has an integral value, hence the left side must also be equal to an integer. Substitute this value of y in the original equation. Assign values to m to find positive integral values of x and y, when m > 1, x becomes negative and when m < 0, y becomes negative; hence we have only two sets of positive integral value for x and y. 2. Solve in positive integers, 5x + 8y Divide by the smaller coefficient 5, = 34. This value of y is not an integral form because the coefficient of y in the fraction If the fraction Зу is multiplied by an integer, it will still be equal to an integer; hence multiply the fraction by a number that will make the coefficient of y one more than a multiple of the denominator. Any value of m > 0 will make x negative and any value of m < 0 will make y negative. Therefore, the only positive integral values 19. A man spent $28 in buying shirts, some at $2 and some at $3. How many of each kind did he buy? 20. A coal dealer sold 10 tons of pea coal, 12 tons of nut coal and 5 tons of egg coal for $245. The next day, at the same price, he sold 8 tons of pea, 5 tons of nut, and 10 tons of egg coal for $204. Find the price of each. |