13. If 5x - y a 10x-11y, and when x = 7, y = 5, find the equa tion between x and y. 14. If the cube of x varies as the square of y, and if x=3 when y = 5, find the equation between x and y. 15. If the square root of a varies as the cube root of b, and if α = 4 when b = 8, find the equation between a and b. 16. If y varies inversely as the square of x, and if y x=3, find x when y = 2. = 8 when 17. If x x y + a, where a is constant, and x = 15 when y = 1, and = 35 when y = 5; find x when y = 2. x= -b, prove that a2 + b2 ∞ ab; and if a ∞ b, prove 19. If y be the sum of three quantities which vary as x, x2, x3 respectively, and when x = 1, y=4, when x = 2, y = 8, and when x = 3, y= 18, express y in terms of x. 20. Given that the area of a circle varies as the square of its radius, and that the area of a circle is 154 square feet when the radius is 7 feet: find the area of a circle whose radius is 10 feet 6 inches. 21. The area of a circle varies as the square of its diameter: prove that the area of a circle whose diameter is 2 inches is equal to the sum of the areas of two circles whose diameters are 1 and 2 inches respectively. 22. The pressure of wind on a plane surface varies jointly as the area of the surface, and the square of the wind's velocity. The pressure on a square foot is 1 pound when the wind is moving at the rate of 15 miles per hour: find the velocity of the wind when the pressure un a square yard is 16 pounds. 23. The value of a silver coin varies directly as the square of its diameter, while its thickness remains the same; it also varies directly as its thickness while its diameter remains the same. Two silver coins have their diameters in the ratio of 4:3. Find the ratio of their thicknesses if the value of the first be four times that of the second. 24. The volume of a circular cylinder varies as the square of the radius of the base when the height is the same, and as the height when the base is the same. The volume is 88 cubic feet when the height is 7 feet, and the radius of the base is 2 feet: what will be the height of a cylinder on a base of radius 9 feet, when the volume is 396 cubic feet? CHAPTER XXXIV. ARITHMETICAL, GEOMETRICAL, AND HARMONICAL PROGRESSIONS. 364. A succession of quantities formed according to some fixed law is called a series. The separate quantities are called terms of the series. ARITHMETICAL PROGRESSION. 365. DEFINITION. Quantities are said to be in Arithmetical Progression when they increase or decrease by a common difference. Thus each of the following series forms an Arithmetical Progression: a, a +d, a +2d, a + 3d, .... The common difference is found by subtracting any term of the series from that which follows it. In the first of the above examples the common difference is 4; in the second it is 6; in the third it is d. 366. The Last, or nth Term, of an A. P. If we examine the series we notice that in any term the coefficient of d is always less by one than the number of the term in the series. Thus the 3d term is a +2d; 6th term is a +5d; 20th term is a +19 d; and, generally, the pth term is a +(p −1)d. If n be the number of terms, and if I denote the last, or nth term, we have 367. The Sum of n Terms in A. P. Let a denote the first term, d the common difference, and n the number of terms. Also let i denote the last term, and S the required sum; then S = a + (a + d)+(a + 2 d) +.....+ (l − 2 d) + (1 − d) +l. and, by writing the series in the reverse order, ... 8=1+(-a)+(l−2d) + ··· + (a + 2 d) + (a + d) + a. Adding together these two series, 28 = (a + 1) + (a + 1) + (a + 1) + ..... to n terms = n (a + 1), 368. In the last article we have three useful formulæ (1), (2), (3); ; in each of these any one of the letters may denote the unknown quantity when the three others are known. Ex. 1. Find the 20th and 35th terms of the series 38, 36, 34, .... Here the common difference is 36 - 38, or - - 2. and the 35th term ... Ex. 3. The first term of a series is 5, the last 45, and the sum 400 find the number of terms, and the common difference. If n be the number of terms, then from (1), 1. Find the 27th and 41st terms in the series 5, 11, 17, ........ Find the last term in the following series: 13. ad, a + d, a + 3d, ... to 30 terms. 14. 2a-b, 4a-3b, 6a - 5b, ... to 40 terms. Find the last term and sum of the following series: Find the number of terms and the common difference when 31. The first term is 3, the last term 90, and the sum 1395. 32. The first term is 79, the last term 7, and the sum 1075. 33. The sum is 24, the first term 9, the last term -6. 34. The sum is 714, the first term 1, the last term 581. 35. The last term is -16, the sum -133, the first term -3. 36. The first term is -75, the sum -740, the last term 1. 37. The first term is a, the last 13 a, and the sum 49 a. 38. The sum is -320 x, the first term 3x, the last term -35 x. 369. If any two terms of an Arithmetical Progression be given, the series can be completely determined; for the data furnish two simultaneous equations, the solution of which will give the first term and the common difference. Ex. Find the series whose 7th and 51st terms are -3 and -355 respectively. If a be the first term, and d the common difference, 370. Arithmetic Mean. When three quantities are in Arithmetical Progression, the middle one is said to be the arithmetic mean of the other two. Thus a is the arithmetic mean between a d and a + d. 371. To find the arithmetic mean between two given quantities. Let a and b be the two quantities; A the arithmetic mear Then, since a, A, b, are in A.P., we must have |