6. If a b c d, then a + ba-bc+d: cd. 7. Given that yox, and that when x = 1, y = the value of y when x = 4. 3, find 8. Given that ∞ ∞ + y, that y∞ x2, and that when x = 1, y = 2, and x = 3, determine the relation between % and x. 9. If s varies directly as y when x is constant, and inversely as a when y is constant, then when x and y both vary ≈ ∞ 10. If x ∞ ∞ + y, u ∞ x - y, and x ∞ u + %, then in general ya. What exception is there to this conclusion? 11. If xxy, and y ∞ x, then Ус x%. 12. If x ∞ y, and y ∞ x, then x + y + x ∞ √ yx + √xx + √xy. 13. A sphere of metal is known to have a hollow space about its centre in the form of a concentric sphere, and its weight is of the weight of a solid sphere of the same substance and radius; compare the inner and outer radii, having given that the weight of a sphere varies as the cube of its radius. 14. There are two vessels, A and B, each containing a mixture of water and wine, A in the ratio of 2: 3, B in the ratio of 3: 7. What quantity must be taken from each, in order to form a third mixture which shall contain 5 gallons of water and 11 of wine? ARITHMETICAL PROGRESSION. 1. The latter half of 2n terms of any arithmetical of the sum of 3n terms of the same series. series = 5. Given the nth and mth terms of an arithmetical series, required the sum of p terms. 6. S1, S2, S3......S are the sums of p arithmetical series continued to n terms; the first terms are 1, 2, 3,...... and the common differences 1, 3, 5,...... Prove that negative value, prove that this value corresponds to a series of n terms, having the common difference d and d for its first term. α 8. There are n arithmetical means between 1 and 31, and the 7th n -1th 5:9; required the number of means. :: 9. Find three numbers in arithmetical progression, whose product 120, and whose sum = 15. = 10. Write down the arithmetical series, the 5th and 9th terms of which are respectively 1 and 9. 11. Determine the relation which must exist between a, b, and c, in order that they may be respectively the pth, gth, and 7th terms of an arithmetical progression. 12. The common difference of 4 numbers in arithmetical progression is 1, and their product 120; find the numbers. 13. Given the nth term of an arithmetical series, and also the sum of n terms; find the series. 14. Prove that 1, 3, 5, 7...... is the only arithmetical progression beginning with unity, in which the sum of the first half of any even number of terms has to the sum of the second half the same constant ratio; and find that ratio. 16. The sums of n terms of two arithmetical series are as 137n: 1 + 3n; find the ratio of their first terms. 17*. Every square number can be represented under the form of an arithmetical progression commencing with unity. 18. In the two series 2, 5, 8,......and 3, 7, 11,....... each continued to 100 terms, how many terms are identical? 19. From two towns, 168 miles distant from each other, two persons, A and B, set out to meet each other; A goes 3 miles the first day, 5 the second, 7 the third, and so on; B goes 4 miles the first day, 6 the second, 8 the third, and so on; in how many days after starting will they meet? 2. Find three numbers in geometrical progression, such that their sum = 14, and the sum of their squares 3. Prove that .1111......= {.3333......}2. = 84. 4. The first term in a geometrical series is 1, and any one term of the series is equal to the sum of all that follow it if continued ad infinitum; required the series. 5. The arithmetical mean between a and b is greater than the geometrical. 6. If quantities are in geometrical progression, their differences are in geometrical progression. 7. There are three numbers in geometrical progression, whose sum = 13, and the sum of the first and second divided by the sum of the second and third = }. Required the numbers. 8. The difference between two numbers is 48, and the arithmetical mean exceeds the geometrical by 18. Find the numbers. 9. In any geometrical series, consisting of an even number of terms, the sum of the odd is to the sum of the even terms as 1 : r, r being the common ratio. 10. Given the sum of three quantities in geometrical progression, and the sum of their reciprocals; find the quantities themselves. 11. In every geometrical progression consisting of an odd number of terms, the sum of the squares of the terms is equal to the sum of all the terms multiplied by the excess of the odd terms above the even. 12. The difference of the means of 4 numbers in geometrical progression is 2, and the difference of the extremes is 7; find the numbers. 13. The second and third terms of a geometrical progression are together equal to 24, and the next two to 216; what is the first? 14*. Find a geometrical series in which the sum of all the terms except the last is equal to the difference between the last and first terms. 15. If S1, S2,......... S be the sums of n geometrical series continued ad infinitum, the first term of which is 1, and the common ratios value of the quantity 16. 1 1 The (p + q)th term of a geometrical series = P, the (p − q)th = Q; find the pth and 9th terms. |