49. A garrison was provisioned for 30 days; after 10 days the garrison was reinforced by 3000 men, and the provisions were then exhausted in 5 days. Find the number of men in garrison at first.

50. A person having to walk 10 miles finds that by increasing his speed a mile an hour he might reach his journey's end 163 minutes sooner than he otherwise would. What time will he take if he quicken his pace halfway?

51. A detachment from an army was marching in regular column, with 7 men more in depth than in front; but the front being increased by 336 men, the detachment was drawn up in 5 lines. Find the number of men.

52. The number of men in both fronts of 2 columns of troops, H and K, when each consisted of as many ranks as it had men in front, was 84; but when the columns changed ground, and H was drawn up with the front K had, and K with the front H had, the number of ranks in both columns was 91. Find the number of men in each column.

53. A certain number of sovereigns, shillings, and sixpences, together amount to £8 6s. 6d., and the amount of the shillings is a guinea less than that of the sovereigns, and a guinea and a half more than that of the sixpences. Find the number of each coin.

54. There is a number composed of 2 figures, of which the figure in the tens' place is 4 times that in the units', and if 54 be subtracted from the number the difference is expressed by the same digits reversed. What is the number?

bows once to each less, the number Find the number of

55. Each one of a company of persons of the others. If the company had been of bows would have been less by 18. persons present.

CXI.

PROGRESSIONS, &c.

1. Define arithmetical progression and harmonical progression. Show that if a2, b2, c2 are in A.P., b+c, c+a, a+b are in H.P.

2. The first term of an arithmetical progression is 2, and the fifth is 18. How many terms must be taken to make the sum 800 ?

3. Sum (1) the arithmetical series 58 +54+... to 30 terms; (2) the geometric series √√...to 5 terms; (3) the series 1+2x+3x2 + ...to n terms.

5. If 3 numbers be in arithmetical progression, and a fourth proportional to them be found, the last 3 terms of the proportion will be in harmonical progression.

6. The first term of a geometric series is 1, and if the series be continued indefinitely, any term is the limit of all the following terms. Find the series.

7. (1) Insert 4 arithmetical means between a and b. Sum the series

(2) 12, 11, 10, to 9 terms;
(3),,, to 6 terms;

(4) √6, √2, √, to infinity.

8. A sets out from a certain place, and travels 1 mile the first day, 2 miles the second day, 3 miles the third day, 4 miles the fourth day, and so on; B sets out 5 days after A, and travels 12 miles a day. How far will B travel before overtaking A?

9. Sum the series

(1) 9+8+7+.........to 50 terms; (2) 1+2+12+...........to 5 terms. 10. Sum the series

(1) 1, 2, 4, &c., to 10 terms;
(2),,, &c., to 6 terms;

(3) 23, 0023, 000023, to infinity.

11. Find the harmonic mean between 2 and 7.

12. Insert (1) 2 arithmetic means, (2) 3 geometric means, between a and b.

....n2=

_n (n+1) (2n+1),

1.2.3

14. Assuming 12+22+32+

find 1+3+6+10+ to n terms.

15. Prove the formula for the sum of a geometrical progression, and apply it in finding the value of the recurring decimal 77777 &c.

16. Find the sum of

(1) 163+143+13+ &c. to 21 terms,

(3) 20, 10, 5, &c., to 6 terms.

18. Insert 5 harmonic means between 1 and 2-1.

21. When are magnitudes in harmonic progression? Find the H.M. between a+b and a−b.

22. If x, y, z, be in geometrical progression, prove that x2y2x2(x¬3+y ̃3+z ̃3)=x3+y3+23.

8-x,

23. When are four quantities said to be in proportion? What value must be given to x to make 1+x, 2+x, 10-x, proportionals?

24. If a b c d, prove the equality

26. If xay: b:: z: c &c., prove that

mx+ny+pz+&c.:ma+nb+pc+&c, :: x: a. 27. If y+z: 36· ›−c:: z+x: 3c-a :: x+y: 3a-b, then x+y+z: ax+by+cz :: a+b+c: a2+b2+c2.

a' b'

28. If and be two unequal fractions, prove that

a+a'

b+b'

is intermediate in value between them.

29. If a b=c:d,

a(a+b+c+d)=(a+b)(c+a).

30. If a b=b: c=cd,

√ab+√bc+√cd=√(a+b+c) (b+c+d).

a с e

b=dƒ
f

a3+b3 b2 (ae+bf) c2e+d2ƒ―d2(a2+b2)

32. If a b=c : d=e: ƒ,

ac ace+bdf+acd=e2: e3 +ƒ3 + e3d.

33. Which is greater

34. Explain fully the meanings of the terms Ratio and Proportion.

a:b=(√a+√c)2 : (√õ+√d)2.

35. Show that a ratio of greater inequality is increased by subtracting the same quantity from both terms.

CXII.

MISCELLANEOUS QUESTIONS.

1. If xy+yz+xz=1, show that

2. Prove that when n is a positive integer "-y" is divisible by x-y, and x2o —y2n by x-y.

3. If a b=c : d=e:f, show that each of these ratios=

√ac+ae+ce: √bd+bf+df="/a"+c"+e":√b"+dn+f".