20. If the numbers a, b, c, d form a G. P., show that

(a — d)2 = (b − c )2 + (c − a)2 + (d − b)2.

21. A merchant's investment yields him each year after the first, three times as much as the preceding year. If his investment paid him $9720 in four years, how much did he realize the first year and the fourth year?

22. On one of the sides of an acute angle a point is taken a feet from the vertex; from this point a perpendicular is let fall on the second side, cutting off b feet from the vertex. From the foot of this perpendicular a perpendicular is let fall on the first side, and from the foot of this perpendicular a third perpendicular is let fall on the second side, and so on indefinitely. Find the sum of all the perpendiculars.

23. Given a square whose side is 2 a. The middle points of its adjacent sides are joined by lines forming a second square inscribed in the first. In the same manner a third square is inscribed in the second, a fourth in the third, and so on indefinitely. Find the sum of the perimeters of all the squares.

§ 4. HARMONICAL PROGRESSION.

1. A Harmonical Progression (H. P.) is a series the reciprocals of whose terms form an arithmetical progression.

Consequently to every harmonical progression there corresponds an arithmetical progression, and vice versa.

2. If three numbers be in harmonical progression, the ratio of the difference between the first and second terms to the difference between the second and third terms is equal to the ratio of the first term to the third term.

Let the three numbers a, b, c be in harmonical progression. By the definition of a harmonical progression,

3. Any term of a harmonical progression is obtained by finding the same term of the corresponding arithmetical progression and taking its reciprocal.

Ex. Find the eleventh term of the harmonical progression 4, 2, 1, ....

The corresponding arithmetical progression is

Therefore the eleventh term of the given progression is

4. No formula has been derived for the sum of n terms of a harmonical progression.

5. A Harmonical Mean between two numbers is a number, in value between the two, which forms with them a harmonical progression.

1

H

E.g., is a harmonical mean between and

음.

Let H stand for the harmonical mean between a and b, then

Ex. Insert a harmonical mean between 2 and 5.

6. Harmonical Means between two numbers are numbers, in value between the two, which form with them a harmonical progression.

E.g., §, 1, 2, 3, are five harmonical means between 3 and 4. Ex. Insert four harmonical means between 1 and 10.

We have first to insert four arithmetical means between 1 and, and obtain

7. To insert n harmonical means between a and b, we insert n arith

1

metical means between and and take their reciprocals. The n

arithmetical means are

a

1

a

a b n+1

a b n+1

(n + 1)ab'

8. Pr. 1. The geometrical mean between two numbers is, and the harmonical mean is . What are the numbers? Let x and y represent the two numbers.

Solving (1) and (2), we obtain x = 1, y = 4, and x = ‡, y = 1.

Pr. 2. If to each of three numbers in geometrical progression the second number be added, the resulting series will form a harmonical progression. What are the numbers?

Let, a, ar represent the three numbers.

We have

=

1

1

a (1+r) gression.

a r

Consequently,+ a, 2 a, ar + a is a harmonical pro

EXERCISES VIII.

Find the last term of each of the following harmonical progressions :

12. Insert 5 harmonical means between 5 and 1.

13. Insert 10 harmonical means between 3 and 4.

a + b

а b

14. Insert 4 harmonical means between - 7 and 1.

15. If x2, y2, z2 be in A. P., prove that y+z, z+x, and x+y are in H. P. 16. If y be the harmonical mean between x and z, prove that

17. The arithmetical mean between two numbers is 6, and the harmonical mean is 35. What are the numbers?

18. If one number exceeds another by two, and if the arithmetical mean exceeds the harmonical mean by, what are the numbers ?

19. The seventh term of a harmonical progression is, and the twelfth term is. What is the twentieth term?

20. The tenth term of a harmonical progression is, and the twentieth term is. What is the first term?

CHAPTER XXVIII.

THE BINOMIAL THEOREM FOR POSITIVE INTEGRAL

EXPONENTS.

1. The expansions of the powers of a binomial, from the first to the sixth inclusive, were given in Ch. VI., § 1, Art. 8, and the laws governing the expansions of these powers were stated.

As yet, however, we cannot infer that these laws hold for the seventh power without multiplying the expansion of the sixth power by a+b; nor for the eighth power without next multiplying the expansion of the seventh power by a+b; and so on.

If, however, we prove that, provided the laws hold for any particular power, they hold for the next higher power, we can infer, without further proof, that because the laws hold for the sixth power, they hold also for the seventh; then that because they hold for the seventh, they hold also for the eighth, and so on to any higher power.

2. If the laws (i.)–(vi.) hold for the rth power, we have ̧r(r−1)(r−2) ar−3z3+***.

(a+b)" = a*+ra"-1b+? (r−1) ar-22 +?

1.2

1.2.3

Notice that only the first four terms of the expansion are written. But it is often necessary to write any term (the kth, say) without having written all the preceding terms.

To derive this term, observe that the following laws hold for each term of the expansion:

(i.) The exponent of b is one less than the number of the term (counting from the left).

=

Thus, in the first term we have b1-1601; in the second, b2−1 = b; in the tenth, b10−1 = 69; and in the kth term, ¿*-1.