19. In an odd number of terms, show that the product of the first and last is equal to the square of the middle term. 20. The product of four terms of a geometrical series is 4, and the fourth term is 4. Find the series.

21. If from a line one third is cut off, then one third of the remainder, and so on, what fraction of the whole will remain when this has been done five times?

22. Of three numbers in geometrical progression, the sum of the first and second exceeds the third by 3, and the sum of the first and third exceeds the second by 21. What

are the numbers?

23. Find two numbers whose sum is 31 and geometrical mean 14.

24. The sum of the squares of two numbers exceeds twice their product by 576; the arithmetical mean of the two numbers exceeds the geometrical mean by 6. Find the numbers.

25. There are four numbers such that the sum of the first and the last is 11, and the sum of the other two is 10. The first three of these four numbers are in arithmetical progression, and the last three are in geometrical progression. Find the numbers.

26. Find three numbers in geometrical progression such that their sum is 13 and the sum of their squares 91.

27. The difference between two numbers is 48, and the arithmetical mean exceeds the geometrical mean by 18. Find the numbers.

28. There are four numbers in geometrical progression, the second of which is less than the fourth by 24, and the sum of the extremes is to the sum of the means as 7 to 3. Find the numbers.

Infinite Geometrical Series.

370. When ris less than 1, the successive terms of a geometrical series become numerically smaller; by taking n large enough we can make the nth term, arn-1, as small as we please, although we cannot make it absolutely zero.

The sum of n terms,

a

1-r

1

rl

1

a

r

by 1

rl

; this sum differs from by taking enough terms we can make l, and consequently

sidered the sum of an infinite number of terms of the series.

1. Find the sum of 1-+-+............

2. Find the value of 0.2363636....

The terms after the first form a decreasing geometrical series in

* Harmonical Progression.

371. A series is called a harmonical series or a harmonical progression when the reciprocals of its terms form an arithmetical series.

Hence, the general representative of such a series is

372. Questions relating to harmonical series should be solved by writing the reciprocals of its terms so as to form an arithmetical series.

373. If a and b denote two numbers, and H their harmonical mean, then, by the definition of a harmonical series,

374. Sometimes it is required to insert several harmonical means between two numbers.

Let it be required to insert three harmonical means between 3 and 18.

Find the three arithmetical means between and T.

These are found to be,, ; therefore, the harmonical means are,,; or 31, 5, 8.

* A harmonical series is so called because musical strings of uniform thickness and tension produce harmony when their lengths are represented by the reciprocals of the natural series of numbers; that is, by the harmonical series 1, 1, 1,,, etc.

EXERCISE 129.

1. Insert four harmonical means between 2 and 12.

2. Find two numbers whose difference is 8 and harmonical mean 14.

3. Find the seventh term of the harmonical series 3,

4. Continue to two terms each way the harmonical series two consecutive terms of which are 15, 16.

5. The first two terms of a harmonical series are 5 and 6. Which term will equal 30?

6. The fifth and ninth terms of a harmonical series are 8 and 12. Find the first four terms.

7. The difference between the arithmetical and harmonical means between two numbers is 14, and one of the numbers is four times the other. Find the numbers.

8. Find the arithmetical, geometrical, and harmonical means between two numbers, a and b; and show that the geometrical mean is a mean proportional between the arithmetical and harmonical means. Also, arrange these means in order of magnitude.

9. The arithmetical mean between two numbers exceeds the geometrical by 13, and the geometrical exceeds the harmonical by 12. What are the numbers?

10. The sum of three terms of a harmonical series is 11, and the sum of their squares is 49. Find the numbers.

11. When a, b, c are in harmonical progression, show that a:c:: a - b : b

- с.

CHAPTER XXIII.

VARIABLES AND LIMITS.

375. Constants and Variables. A number that, under the conditions of the problem into which it enters, may be made to assume any one of an unlimited number of values is called a variable.

A number that, under the conditions of the problem into which it enters, has a fixed value is called a constant.

Variables are represented by x, y, z; constants by a, b, c, and by the Arabic numerals.

376. Limits. When the value of a variable, measured at a series of definite intervals, can by continuing the series be made to differ from a given constant by less than any assigned quantity, however small, but cannot be made absolutely equal to the constant, the constant is called the limit of the variable, and the variable is said to approach indefinitely to its limit.

Consider the repetend 0.333....., which may be written LO + TO + Tổ명이 +

The value of each fraction after the first is one tenth of the preceding fraction, and by continuing the series we shall reach a fraction less than any assigned value, however small; that is, the values of the successive fractions approach O as a limit.

The sum of these fractions will always be less than; but the more terms we take, the nearer does the sum approach as a limit.