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9. If a and b be prime numbers, the number of numbers prime to a b and less than a b, is equal to
(a – 1) (6 – 1), unity being considered as one of them.
10. Given the logarithm of n, to find the logarithm of n + 1.
1. A person paid a tax of 10 per cent upon his income ; what must his income have been, when, after he had paid the tax, he had £1250 remaining ?
12 The difference between any number and that number inverted, is divisible by 9. 13. Prove that
u2 23 44 log (1 + u)
2 3 4.
may be computed.
17. Find two numbers such that their sum, product, and the difference of their squares may be all equal.
18. How many different ways may £100 be paid in crowns and guineas ?
19. In the expansion of (a + b + c + &c.)", where w=p +9+5 + &c.; find the coefficient of the term involving ap.69.ch. &c.
20. Apply the duodenary scale of notation to find the solidity of a cube, the side of which is 13ft. 7in. 7 pts.
21. Find a series of fractions converging to v 17.
22. The first term of a geometric series continued in infinitum is 1, and any term is equal to the sum of all the succeeding terms. Required the series.
23. Given the sum of 2n quantities in arithmetical progression and the sum of their squares, to find the quantities themselves.
24. Shew that yun – 1 is divisible by either of the quantities y - 1 and y" – 1 without a remainder.
25. Given the mth and nth terms of an harmonical progression, to find the (m + n)th term. 26. Prove that
(2n + 1)2 lo
(2n + 1)2 -- 1 27. A and B can do a piece of work in m days; B and C in n days: in what time can A and C do the same, it being supposed that A can do p times as much as B in a given time?
28. Prove that the cube of any number and the number itself, being divided by 6, leave the same remainder.
29. Find the present worth of £P due n years hence, at r per cent. discount.
30. Investigate the rule for the extraction of the square root in whole numbers ; and determine, generally, the limit which the remainder after any operation cannot exceed.
31. Prove the rule for single position : state to what limitations it is subject; and apply it to find such a number that when divided by 3, 4, and 5, respectively, the sum of the quotients may be 94.
32. Find a quantity, which when multiplied into a3 -— 63 renders the product rational.
33. Four persons, A, B, C, D, in order, cut a pack of cards, replacing them after each cut, on condition that the first who cuts a heart shall win. What are their respective probabilities of success?
34. A given annuity which is to continue 3 n years, is left equally between A and B; A receives the whole for n years, and B the whole for the remainder of the time; it is required to find the present worth of the annuity, and the rate of compound interest.
35. Given the sum of three quantities in geometrical progression, and the sum of their reciprocals, to find the quantities themselves.
36. In what time will the amount of £P at r per cent. simple interest be equal to p times the interest of the same sum, and what is the rate per cent. when the required time is
37. Of the two quantities a® +a%b2 + a264 +66 and (a3 +63)?, shew which is the greater.
38. The sum of a series of quantities in geometrical progression wanting the first term, is equal to the sum of all the terms except the last, multiplied by the common ratio. Required a proof.
39. Prove that (Aa + Bb + Cc + .)2 = (A + B +C + .......) (Aa? + B62 + Cc + .) - AB (a - b)
AC (a -- c)2 – BC (b − c)2 – .
40. A and B are at play together, and the latter having lost p stakes, is determined to play till he has won them again; find the probability that this never takes place, supposing the play to continue without limitation ; his number of chances (6) for winning any assigned game being less than (a) that for the contrary.
41. Find the greatest term of the expansion of (a + b)".
42. 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.
B 43. Find the sum of 1 +
+ a, ß, y, .. being the coefficients of the expansion of (a+b)”.
44. There are four numbers, the first three of which are in arithmetical, and the last three in harmonical progression; it is required to prove that the first has to the second the same ratio which the third has to the fourth.
45. Extract the fourth root of mo (m2-3n2) + n°(n2-3m) + 4(m-n) (m + nmn - 1.
46. Find three fractions having different prime denominators, whose sum shall be l z.
47. The area of a floor is 403ft. 7in. and length 27ft. 10in. 1824 find the width by duodecimals.
48. If a be greater than b, an – B1 is greater than
A' B' C 50. Let
.... be the 1st, 2d, 3d, .... approxi-
A 1 1 1 1
+ AB BC CD DE
51. From a bag containing two balls, a white ball is drawn twice following, the ball having been replaced in the bag after the first drawing; required the probability that both balls are white, and that the ball being a second time replaced, a white ball will be drawn at the third trial. 52. Prove that log x =n
(x - 1) nearly, when n is very great.
53. If N = nth term of the expansion of ar, determine n when the series reckoned from that term begins to converge ; and shew that the sum of all the terms which follow N is less
1 - x log a 54. Transform 8978 from a local value 11, and 3256 from a local value 7, to a system in which the local value is 12; and multiply the numbers together in that system.
55. Prove that (1 + x)" may in all cases be expressed by a series of the form
1 + ax + 6x2 + A 56. If
be a fraction in its lowest terms, B greater than A, and of the form B’.21.51; the quotient will be a mixed cir
culating decimal, and the higher of the indices m, n will be the number of figures in the part which does not recur.
57. Represent a million acording to the duodenary scale of notation.
58. Divide 292 into two other square integer numbers.
59. Required the present worth of £75. due 15 months hence at 5 per cent. per annum.
60. Shew that if the sum of the digits in the odd places be subtracted from any number expressed in decimal notation, and the sum of the digits in the even places be added to the same number, the result is divisible by 11.
61. The difference of the means of four numbers in geometrical progression is 2 and the difference of the extremes is 7. Required the numbers.
62. The present value of an annunity of £1. to continue x years is £10. and the present value of an annuity of £1. to continue 2x years is £16. What is the rate of interest ?
63. Explain the principal advantages of Briggs' system of logarithms.
64. What is the present value of a freehold estate of £150. a year, allowing the purchaser 6 per cent. compound interest ?
65. Shew that the greater two consecutive numbers are, the less is the difference between their logarithms.
66. The sum of two numbers is 6, and the sum of their cubes 72. Required the numbers.
67. Upon a given straight line as an hypothenuse, describe a right-angled triangle which shall have its three sides in continued proportion.
41 68. Find a series of fractions converging to
72 69. Sum the following series :
+1+ .. to 8 terms. 4