EXAMPLES. x+y:x::5:3 1. Given {*+;:;5:3} to find x and y. Since {t xy=6 by division, Art. 305, Hence Substituting this value of y in the second equation, we ob xy=63 Substituting this value of x in the second equation, we find y=2, and x=4. 3. Given { (x+3)2 : (x—3)2 : : 64 : 1 : 1 } to find x and y. Substituting this value of x in the second equation, we find From the first equation, by division, Art. 305, Hence or 3xy (x-y): (x-y)3:: 60: 1. 960: (x-y)2:: 60:1, 16: (x-y)2::1:1. 5. Given {x3-y? x+y=6 Ans. { x=4 or 2. y=2 or 4. y-x+Vα-x: √α-x:: 5:2 } to find x and y. 7. Given x+√x: x−√x : : 3 √x+6:2√x to find x. 8. What number is that to which if 1, 5, and 13 be severally added, the first sum shall be to the second as the second to the third? Ans. 3. 9. What number is that to which if a, b, and c be severally added, the first sum shall be to the second as the second to the third? 12-ac Ans. a-2b+c 10. What two numbers are as 2 to 3, to each of which if 4 be added, the sums will be as 5 to 7? 11. What two numbers are as m to n, to each of which if a be added, the sums will be as p to q? Ans. am (p—9), an (p−q) mq-np mq-np 12. What two numbers are those whose difference, sum, and product are as the numbers 2, 3, and 5 respectively? Ans. 2 and 10. 13. What two numbers are those whose difference, sum, and product are as the numbers m, n, and p? 2p 2p Ans. and 14. Find two numbers, the greater of which shall be to the less as their sum to 42, and as their difference to 6. Ans. 32 and 24. 15. Find two numbers, the greater of which shall be to the less as their sum to a and their difference to b. 16. There are two numbers which are in the ratio of 3 to 2, the difference of whose fourth powers is to the sum of their cubes as 26 to 7. Required the numbers. Ans. 6 and 4. 17. What two numbers are in the ratio of m to n, the difference of whose fourth powers is to the sum of their cubes as p to q? Ans. mp. m3+n3 np and X Չ 18. Two circular metallic plates, each an inch thick, whose diameters are 6 and 8 inches respectively, are melted and formed into a single circular plate 1 inch thick. Find its diameter, admitting that the area of a circle varies as the square of its diameter. 19. Find the radius of a sphere whose volume is equal to the sum of the volumes of three spheres whose radii are 3, 4, and 5 inches, admitting that the volume of a sphere varies as the cube of its radius. 20. Find the radius of a sphere whose volume is equal to the sum of the volumes of three spheres whose radii are r, r', and r". CHAPTER XVI. PROGRESSION S. ARITHMETICAL PROGRESSION. 320. An arithmetical progression is a series of quantities which increase or decrease by a common difference. Thus the following series are in arithmetical progression: 1, 3, 5, 7, 9,... 20, 17, 14, 11, 8, ... a, a+d, a +2d, a+3d,... In the first example the common difference is +2, and the series forms an increasing arithmetical progression; in the second example the common difference is 3, and the series forms a decreasing arithmetical progression. In the third example the common difference is +d, and in the fourth example it is -d. 321. In an arithmetical progression having a finite number of terms, there are five quantities to be considered, viz., the first term, the last term, the number of terms, the common difference, and the sum of the terms. When any three of them arc given, the other two may be found. We will denote the first term by the last term by the number of terms by the common difference by and the sum of the terms by The first term and the last term are called the extremes, and all the other terms are called arithmetical means. 322. In an arithmetical progression the last term is equal to the first term plus the product of the common difference by the number of terms less one. Let the terms of the series be represented by a, a+d, a+2d, a+3d, a+4d, etc. Since the coefficient of d in the second term is 1, in the third term 2, in the fourth term 3, and so on, the nth term of the series will be in which d is positive or negative according as the series is an increasing or a decreasing one. 323. The sum of any number of terms in arithmetical progres sion is equal to one half the sum of the two extremes multiplied by the number of terms. The term preceding the last will be 7-d, the term preceding that 1-2d, and so on. If the terms of the series be written in the reverse order, the sum will be the same as when written in the direct order. Hence we have +a. s=a+(a+d)+(a+2d)+(a+3d)+.... +l, s=l+(l−d)+(l − 2d) + (l −3d)+.... Adding these equations term by term, we have 2s=(a+1)+(a+1)+(a+1)+.... +(a+1). Here a+ is taken n times; hence 324. In an arithmetical progression the sum of the extremes is equal to the sum of any two terms equidistant from the extremes. This principle follows from the preceding demonstration. It may also be shown independently as follows: The mth term from the beginning is a+(m-1)d. The mth term from the end is And the sum of these terms is a+l l—(m—1)d. 325. To insert any number of arithmetical means between two given terms. The whole number of terms in the series consists of the two |