Now (a/b) is always positive, and therefore greater than zero, unless a = b.

Since the arithmetic mean of two positive quantities is greater than their geometric mean, it follows from Art. 227 that the geometric mean is greater than the harmonic.

229. To insert n harmonic means between any two quantities a and b.

reciprocals of these will be the required harmonic means. The arithmetic means are

Hence, by simplifying these terms and inverting them, the required harmonic means will be found to be

230. It is of importance to notice that no formula can be found which will give the sum of any number of terms in harmonical progression.

EXAMPLES XXI.

1. Shew that, if a, b, c be in a. P., then will a2 (b + c), b2 (c+a), c2 (a + b) be in A. P.

2. Find four numbers in A. P. such that the sum of their squares shall be 120, and that the product of the first and last shall be less than the product of the other two by 8.

3. If a, b, c be in A. P., and b, c, d be in H. P., then will a b c d.

4.

=

Find three numbers in G. P. such that their sum is 14, and the sum of their squares 84.

5. If a, b, c be in arithmetical progression, and x be the geometric mean of a and b, and y be the geometric mean of b and c; then will a, b, y2 be in arithmetical progression.

will

6. Shew that, if a, b, c be in harmonical progression, then

b+c− a' c + a

progression.

7. Shew that, if a, b, c, d be in harmonical progression, then will

8. Shew that, if a, b, c be in harmonical progression,

9. Shew that, if a, b, c be in H. P., then will

10. If a, b, c be in A. P., b, c, d in G. P., and c, d, e in H. P.;

13. If x, a1, α, y be in A. P., x, 91, 9,, y in G. P., and x, h,, h2, y in н. P., then

14. The sum of the first, second, and third terms of a G. P. is to the sum of the third, fourth and fifth terms as 1 : 4, and the seventh term is 384.

Find the series.

15. If a1, α, a,......ɑ, be in harmonical progression, prove that a1a2 + + α-α = (n − 1) a ̧α„·

2 3

......

16. If a, x, y, b be in arithmetical progression, and a, u, v, b be in harmonical progression, then xv =

yu = ab.

17. Three numbers are in arithmetic progression, and the product of the extremes is 5 times the mean; also the sum of the two largest is 8 times the least. Find the numbers.

= b = c.

If a, b, c, be in A. P., and a3, b3, c2 be in H. P., prove

b, c are in G. P., or else a =

20. If x be any term of the arithmetical progression and y be the corresponding term of the harmonical progression whose first two terms are a, b, then will x ·ay-a::b: y.

21.

Shew that, if a be the arithmetic mean between 6 and c, and b be the geometric mean between a and c, then will c be the harmonic mean between a and b.

Prove that

22. The series of natural numbers is divided into groups as follows: 1; 2, 3; 4, 5, 6; 7, 8, 9, 10; and so on. the sum of the numbers in the kth group is k (k2 + 1).

23. An A. P. and an H. P. have each the first term a, the same last term l, and the same number of terms n; prove that the product of the (r+1)th term of the one series and the (n-rth term of the other is independent of r.

24. Terms equidistant from a given term of an A. P. are multiplied together; shew that the differences of the successive terms of the series so formed are in A. P.

2n' 3n

25. Shew that, if S, S, S be the sum of n terms, of 2n terms, and of 3n terms respectively of any G. P., then will S (San-S2n) = (San — S2)3.

26.

2n

If a, b, c be either in A. P., in G. P., or in H. P., and n be any positive integer, then a" + c" > 2b".

27.

If P, Q, R be respectively the pth, qth, and 7th terms (i) of an A. P., (ii) of a G. P., and (iii) of an H. P., then will

(i) P(q-r) + Q (r− p) + R (p − q) = 0,

(ii) Pa-r. Qr-o . Ro–o = 1,

-

(iii) QR (q-r) + RP (r − p) + PQ (p − q) = 0.

29.

Shew that, if a1, α2, ɑ..................., ɑn be all real, and if

2

2

(a + a + ......+ a3n_‚) (a ̧3 + a ̧3 +........ + an3)

= (α ̧à ̧ + α ̧a ̧+...... + A2_, α 2)3,

then will a1, α, a....... be in G. P.

2-1

30. Shew that any even square, (2n)3, is equal to the sum of n terms of one series of integers in A. P., and that any odd square, (2n + 1), is equal to the sum of n terms of another A. P. increased by unity.

31. Prove that any positive integral power (except the first) of any positive integer, p, is the sum of p consecutive terms of the series 1, 3, 5, 7, &c.; and find the first of the p terms when the sum is p".

32. If an A. P. and a G. P. have the same first term and the same second term, every other term of the A. P. will be less than the corresponding term of the G. P., the terms being all positive.

CHAPTER XVIII.

SYSTEMS OF NUMERATION.

231. IN arithmetic any number whatever is represented by one or more of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, called figures or digits, by means of the convention that every figure placed to the left of another represents ten times as much as if it were in the place of that other. The cipher, 0, which stands for nothing, is necessary because one or more of the denominations, units, tens, hundreds, &c., may be wanting.

The above mode of representing numbers is called the common scale of notation, and 10 is said to be the radix or base.

232. Instead of ten any other number might be used as the base of a system of numeration, or as it is generally called the radix of a scale of notation; and to express a number, N, in the scale whose radix is r, is to write the number in the form ...... ddd,do, where each of the digits do, d, d, d....... is less than r, and where d, stands for d units, d, stands for d1 × r, d, for d2 × r2, and so on.

Note. Throughout this chapter each letter stands for a positive integer, unless the contrary is stated.