x2 x3

62. If y=x- + &c., show how to determine x.

2 3

63. If the sum of the nth and 2nth terms of a geometrical progression be given, and also the sum of the 2nth and 3nth terms, find the first term and the common ratio.

64. Sum the series of which 2r+3+2 x 3" is the rth term.

65. Find the present value of an annuity of £A to be continued for n years at r per cent. compound interest.

66. If the annuity become a perpetuity, find the number of years' purchase that must be paid for it at 5 per cent.

67. Show how to find the number of shot in a complete rectangular pile; and apply the result to find the number of shot in an incomplete pile having 200 and 160 shot in the sides of its base, 70 and 30 in the sides of its top course.

68. The product of 5 numbers in A.P. is 945, and their sum is 25; find the numbers.

69. Expand (a2-x2)-2 to 6 terms by the binomial theorem. 70. If four magnitudes he proportionals, the sum of the greatest and least is greater than the sum of the other two.

71. The number of combinations of n things, taken 4 together, is to the number taken 2 together as 51 to 2. Find n.

72. Write down all the numbers that can be composed of the four digits 3, 4, 5, 6, which are divisible by 11.

73. Six papers are to be set in an examination, two of them in mathematics; in how many different orders may the papers be given, provided only that the two mathematical papers do not come together?

74. Find a third proportional to

2a2b2

a2+b2

and ab.

75. Prove that the coefficient of x2n in the expansion of

76. The difference between the numbers of shot in the two sides of the base of an incomplete rectangular pile is 7, and the number in the longer side of the top course is 15. How many shot are required to complete the pile?

77. A dealer having laid in a stock of a certain article, began to sell it by retail. The first day he made a profit of 3d., the second of 4.2d., and so on, the profit increasing by 1.2d. a day, until the stock was disposed of; he then found that he had realised a profit of 14s. 3d. How many days did he continue to sell the article?

78. Prove that the difference of the

odd numbers is divisible by 8.

squares of any two

79. Expand 11 in the form of a continued fraction. 80. If n be a very large number, prove that log (n + 1) −

log n is very nearly equal to

system of logarithms.

M

n

M being the modulus of the

81. Write down the first five terms of the expansion of (1+x)2o, and show, when n is a whole number, that the sum of the coefficients of the expansion=(2"). Also assuming the (p+1)th coefficient of the expansion equal to the (p+3)th coefficient, prove p=n−1.

7.

S (·21)*×(·21)*× (·21)a×3652]}

(00416)2 × (3125)3 / 365

8. (34)3× (24·68)3.

9. A fourth proportional to 254·5, 4000·6, ·028.

12. A mean proportional to √4·756 and 3/00782.

24. A fourth proportional to 1·027, 3⁄4/2·546, and

(31.027)2.

25. A third proportional to 1.6 and

3/44

794.81

26. A fourth proportional to 8.37, (·84)2, 5/ ·054321.

27. The sixth root of '000000004096.

The 5th term of 5, 15, 45...

28.

30. A mean proportional between

(2·0736) and (000000262144).

35. A fourth proportional to 41·06}, '000562 and 120%.

1. Define the terms 'Logarithm,' 'Base,' 'Mantissa,' 'Characteristic,' 'Modulus.'

2. Prove that log (35 x 46)=5 log 3+6 log 4.

3. Prove that log10N=log,Nx.

1

log. 10.

4. Explain the method of finding the characteristic of log10 '0005.

5. What are the logs of 10, 100, 001 to the base 10; and of 1000 to the base '01 ?

6. Given log 2=30103, log 18=1·2552725, log 21= 1-3222193: find (1) log 0075, (2) log 31·5, (3) log 128, (4) log 125, (5) log 2500.

7. In any system, what are the logs (1) of the base, (2) of 1, (3) of O?

8. Prove that in the common system of logs the log of a number having n digits in the integral part is n-1; and that the characteristic of the log of a decimal having n ciphers between the point and the first significant digit is -(n+1). Also show that the same mantissa serves for all numbers consisting of the same digits in the same order. 9. Given log 2=30103, log 3=4771213, log 7='8450980, find (1) log 60, (2) log 03, (3) log 105.

10. Given log 1752=3.2435341, log 1752·1=3·2435589, construct a table of proportional parts, and find

log 17.520-87. Find without tables log101000. 11. Divide 14.326847 by 9.

12. What is the logarithm of 813/3 to the base 3√3? How would you transform a system of logs from the base 8 to the base 4? What advantages belong to a system with 10 at the base?

13. Prove that log,axlog,b=1. Can the same number have more than one logarithm, real or imaginary? If log 25-6, find the base.

14. Prove that log10N=log105 x log,N, and find without tables log258, having given log102=30103.

15. Given log102=30103, find log10125, and the log10 of the mean proportional between

16. Of what number is -5 the log to the base 10? What is the log of 256 to the base 2? If log102=30103, and log107=-845098, find log10 C035; log10 N=1.5, find N without tables.

17. If log10125=2-09691, find log102. Find without