30. The geometrical mean of two numbers is 4 and their sum is 10. Find the numbers.

31. The fourth term of a G.P. is 192, the seventh term is 12,288. Find the first term and the ratio.

32. If the same number be added to or subtracted from each term of a G.P., is the resulting series geometrical?

33. The product of the first and last of four numbers in G.P. is 64. Their quotient is also 64. Find the numbers.

34. The product of four numbers in G.P. is 81. The sum of the second and third terms is. Find the numbers.

35. If every term of a G.P. be multiplied by the same number m, is the resulting series a G.P.? If so, what are the elements?

36. The sum of three numbers in G.P. is 42. The difference between the squares of the first and the second is 60. What are the numbers?

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

38. Four numbers are in G. P. The difference between the first and the second is 4, the difference between the third and the fourth is 36. Find the numbers.

39. A ball falling from a height of 60 feet rebounds after each fall one third of the last descent. What distance has it passed over when it strikes the ground for the eighth time?

40. The difference between the first and the last of three terms in G.P. is four times the difference between the first and second terms. The sum of the numbers is 208. Find the numbers.

41. An invalid on a certain day was able to take a single step of 18 inches. If he was each day to walk twice as far as on the preceding day, how long before he can take a five-mile walk?

42. The difference between the first and the last of four numbers in G.P. is thirteen times the difference between the second and third terms. The product of the second and third terms is 3. Find the numbers.

137. Infinite series. When the number of terms of a G.P. is unlimited it is called an infinite geometrical series.

In the series a, ar, ar2, ..., when r> 1, evidently each term is larger than the preceding term. The series is then called increasing. When < 1, each term is smaller than the preceding term and the series is called decreasing.

When r>1, evidently becomes very large for large values of n. For this case, then, the sum of the first n terms becomes very large for large values of n. In fact we can take enough terms so that s will exceed any number we may choose. If, however, r< 1, as n increases in value " becomes smaller and smaller. In fact we can choose n large enough so that " is as small as we wish, or as we say, approaches 0 as a limit. But since may be made as small as we wish, ar also approaches 0 as a limit, and conseapproaches 0 as a limit. Thus when r<1 the

quently

азы 1 value of the sum of the first n terms approaches

r

becomes very great. This we express in other words by asserting that the sum of the infinite series

10. How large a value of n must one take so that the sum of the first

n terms of the following series differs from the sum to infinity by not more than .001?

11. What is the value of the following recurring decimal fractions?

(a) .212121....

Solution: This decimal may be written in the form

ADVANCED ALGEBRA

CHAPTER XV

PERMUTATIONS AND COMBINATIONS

138. Introduction. Before dealing directly with the subject of the chapter we must answer the question, In how many distinct ways may two successive acts be performed if the first may be performed in p ways and the second may be performed in q ways? Suppose for example that I can leave a certain house by any one of four doors, and can enter another house by any one of five doors, in how many ways can I pass from one house to the other? If I leave the first house by a certain door, I have the choice of all five doors by which to enter the second house. Since, however, I might have left the first by any one of its four doors, there are 4.5 = 20 ways in which I may pass from one house to the other. This leads to the

THEOREM. If a certain act may be performed in p ways, and if after this act is completed a second act may be performed in q ways, then the total number of ways in which the two acts may be performed is p·q.

With each of the p possible ways of performing the first act correspond q ways of performing the second act. Thus with all the p possible ways of performing the first act must correspond p times as many ways of performing the second act. That is, the two acts may be performed in p q ways.

It is of course assumed in this theorem that the performance of the second act is entirely independent of the way in which the first act is performed.

EXERCISES

1. I have four coats and five hats. How many different combinations of coat and hat can I wear?

Solution: The first act consists in putting on one of my coats, which may be done in four ways; the second act consists in putting on one of my hats, which may be done in five ways. Thus I have 4.5 = 20 different combinations of coat and hat.

2. In how many ways may the two children of a family be assigned to five rooms if they each occupy a separate room?

3. A gentleman has four coats, six vests, and eight pairs of trousers. In how many different ways can he dress?

4. I can sail across a lake in any one of four sailboats and row back in any one of fifteen rowboats. In how many ways can I make the trip?

5. Two men wish to stop at a town where there are six hotels but do not wish quarters at the same hotel. In how many ways may they select hotels?

6. A man is to sail for England on a steamship line that runs ten boats on the route, and return on a line that runs only six. In how many different ways can he make the trip?

7. In walking from A to B one may follow any one of three roads; in going on from B to C one has a choice of five roads. In how many different ways can one walk from A to C ?

139. Permutations. Each different arrangement either of all or of a part of a number of things is called a permutation. Thus the digits 1, 2 have two possible permutations, taken both at a time, namely, 12 and 21.

The digits 1, 2, 3 have six different permutations when two are taken at a time, namely, 12, 13, 21, 23, 31, 32. For if we take 1 for the first place, we have a choice of 2 and 3 for the second place, and we get 12 and 13. If 2 is in the first place, we get 21 and 23. Similarly, we get 31 and 32. In this process it is noted that we can fill the first of the two places in any one of three ways; the second place can be filled in each case in only two ways. Thus by the Theorem, § 138, we should expect 3.26 permutations of three things taken two at a time. We observe that this product 3.2 has as its first factor 3, which is the total number of things considered. The number of factors is equal to the number of digits taken at a time, i.e. two. This leads to the general