Ex. 45.

5. Show that a−x: a +x is> or <a2 — x2 : a2 + x2 according as ax is a ratio of less or greater inequality.

7. Find the ratio compounded of a +x: a-x, a2 + x2 : (a + x)2, and (a2x2)2: aa—x4.

8. If a:b>c: d; show that a+c: b+d<a: b, but >c: d.

9. If a: b=c:d; show that 7a+b: 3a+56=7c+d: 3c+5d.

10. If a be the greatest of the 4 proportionals a, b, c, d; show that a+db+c; and that an+d">b" + cn.

ž.

14. If y2xx2 — a2, and y=22, when x= (a2 + b2) §.

a

15. If y❜xx, and y=+2a, when x=a.

16. If xay and yaz; show that

(ax+by+cz)α {h(xy) +k(xz) +l(yz) ž}.

17. 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 II of wine?

18. The value of diamonds a as the square of their weights, and the square of the value of rubies a as the cube of their weights; a diamond of a carats is worth m times a ruby of b carats, and both together are worth £c: find the values of a diamond and ruby, each weighing a carats.

ARITHMETICAL PROGRESSION.

If S be the sum of n terms of a series in Ar. Prog., a, 1 the 1st and nth terms respectively, d the common difference

1=a+ (n-1)d;

Ex. 46.

S={za+(n−1)d}.

1. Find the 15th term of the series 3, 7, 11, &c.

E

Ex. 46.

2. Find the IIth term of the series 5, 1, 3, &c. 3. Find the 20th term of the series 57, 54, 51, &c. 4. Find the 8th term of the series,,, &c.

5. Find the 19th term of the series, 14, 23, &c.

6. Of 1+3+5+&c. to 20 terms.

7. Of 2+7+12+ &c. to 101 terms. 8. Of 12+7+2-3-&c. to 9 terms. 9. Of −5−1+3+&c. to 12 terms. 10. Of 2+24+2+&c. to 12 terms. 11. Of 13+12+11+&c. to 40 terms. 12. Of 1+23+4+&c. to 22 terms. 13. Of 2+3+4+ &c. to 5 terms.

14. Of 1+++ &c. to 15 terms.

15. Of 6++5+ &c. to 25 terms.

16. Of 17+9+15+ &c. to 51 terms.

17. Of —7—5—4—&c. to 21 terms.

18. Of --- &c. to n terms.

19. Of 23+2+++&c. to n terms.

21. Of (a+x)2 + (a2+x2) + (a−x)2 + &c. to n terms.

a-b 3a-2h 5a-3b
a+b a+b a+b

22. Of +

+

+&c. to n terms.

How many terms of the series—

23. 7, 9, 11, &c. amount to 40?

24. 19, 17, 15, &c. amount to 91 ?

25. 54, 51, 48, &c. amount to 513?

26. 034, 0344, 0348, &c. amount to 2.748?

27. The first term of an AR. series is, the common difference 13,

and the sum of the series 22; find the number of terms.

Ex. 46.

28. The first term of an AR. series is 5, the number of terms is 30, and their sum 1455; find the common difference.

29. The first term of an AR. series is 19, the last term, and the nnmber of terms 12; find the common difference.

30. The sum of II terms of an AR. series is 22, and the common difference is ; find the first term.

31. The first term of an AR. series is 17, the last term - 123, and the sum 257; find the common difference.

32. Insert 3 AR. means between 117 and 477.

33. Insert 4 AR. means between 3 and 18.

34. Insert 4 AR. means between 2 and -18. 35. Insert 9 AR. means between 1 and

I.

36. Insert 7 AR. means between -14 and 44.

37. There are n AR. means between 1 and 31, such that the 7th mean: (n-1)th mean=5:9; required n.

38. The sum of n AR. means between 1 and 19: sum of the first n-2 of them :: 5:3; required n.

39. There are n AR. means between a and b, and between the pth and qth terms of these means there are r AR. means inserted; find the mth term of the last set.

40. The 5th and 9th terms of an AR. series are 13 and 25 respectively; what is the 7th term ?

41. The nth term of an AR. series is (3n-1); find the first term, common difference, and the sum of n terms.

42. The sums of 2 AR. series each to n terms, are as 13-7n : 1+ 3n; find the ratio of their first terms, and also of their second

terms.

43. In the series 1, 3, 5, &c. the sum of 2r terms: the sum of r terms::: I; determine the value of x.

44. Find the ratio of the latter half of 2n terms of any AR. series, to the sum of 3n terms of the same series.

45. If m and n be the (p+q)th and (p-q)th terms respectively of an AR. series; find the pth and qth terms.

46. If a, b and c be the pth, qth and rth terms respectively of an AR. series; show that a(q-r)+b(r−p)+c(p−q)=0.

Ex, 46.

47. If s1, S2, S3,

s, be the sums of r AR. series, each continued to n terms; I, 2, 3, ..r their first terms, and 1, 3, 5, (2r-I) their common differences respectively; required the sum of the series s,+2+3+...+8r.

48. If Sn, Sn+1, S2+2. . . denote the sums of n, n + 1, n + 2, . . . terms of an AR. series; find the sum of sn + Sn+1, + Sn + 2 + .. to n

terms.

...

49. In the two series 2, 5, 8, . . . and 3, 7, II, each continued to 100 terms; find how many terms are common to both series.

50. A debt can be discharged in a year by paying Is. the first week, 3s. the second, 5s. the third, and so on; required the amount of the debt and the last payment.

51. From two towns 168 miles distant, A and B set out to meet each other; A went 3 miles the first day, 5 the second, 7 the third, and so on; B went 4 miles the first day, 6 the second, 8 the third, and so on in how many days did they meet?

52. Find 4 numbers in AR. progression, such that the sum of the squares of the first and second be 29; and the sum of the squares of the third and fourth be 185.

53. Given P and Q the mth and nth terms of an AR. series; required the rth term.

Ex. 47. Find the sum

1. Of 1+2+3+ &c. to 15 terms. 2. Of 12+32+52+&c. to 21 terms. 3. Of 22+52+82+&c. to n terms.

4. Of 13 +23+33 +&c. to n terms.

5. Of 1.2+2.3+3.4+4.5+&c. to 10 terms. 6. Of 1.3+3.5+5.7+7.9+&c. to 12 terms. 7. Of 3.8+6.11+9.14+&c. to n terms.

8. Of 2.5-4.7+6.9-&c. to 2r terms.

9. Of the triangular numbers 1, 3, 6, 10, 15, ... to n terms. 10. Of the pyramidal numbers 1, 4, 10, 20, 35, ... to n terms. 11. Given n2 +(n + 1)2 + (n + 2)2 +..to 9 terms=501; required n. 12. Determine the AR. series of II terms, whose sum is 220, and the sum of their cubes 147400.

GEOMETRICAL PROGRESSION.

If S be the sum of n terms of a series in Geom. Prog., a, z the 1st and nth terms respectively, r the common ratio; then will

If the series be convergent, in which case r<1; and be extended without limit so that n becomes oo; then will the limiting sum

1. Find the 5th term of the series 5, 10, 20, &c.
2. Find the 7th term of the series 54, 27, 131, &c.
3. Find the 6th term of the series 31, 24, 14, &c.
4. Find the 5th term of the series 27, -45, 75, &c.
5. Find the 7th term of the series −21, 14, -97, &c.

Find the limit of the sum of the following infinite series: