had traveled as many days as he went miles in a day, he met A. Required the distance from C to D.

Ans. 76 or 152 miles; both numbers will answer the condition.

19. A farmer received 24 dollars for a certain quantity of wheat, and an equal sum at a price 25 cents less by the bushel for a quantity of barley, which exceeded the quantity of wheat by 16 bushels. How many bushels were there of each? Ans. 32 bushels of wheat, and 48 of barley.

20. A laborer dug two trenches, one of which was 6 yards longer than the other, for 17 pounds, 16 shillings, and the digging of each of them cost as many shillings per yard as there were yards in its length. What was the length of each? Ans. 10 and 16 yards.

21. A and B set out from two towns which were distant from each other 247 miles, and traveled the direct road till they met. A went 9 miles a day, and the number of days at the end of which they met, was greater, by 3, than the number of miles which B went in a day. How many miles did each travel? Ans. A, 117, and B 130 miles. 22. The fore wheels of a carriage make 6 revolutions more than the hind wheels, in going 120 yards; but if the circumference of each wheel be increased 1 yard, they will make only 4 revolutions more than the hind wheels, in the same distance; required the circumference of each wheel.

Ans. 4 and 5 yards.

23. There are two numbers whose product is 120. If 2 be added to the lesser, and 3 subtracted from the greater, the product of the sum and remainder will also be 120. What are the numbers ? Ans. 15 and 8.

24. There are two numbers, the sum of whose squares exceeds twice their product, by 4, and the difference of their squares exceeds half their product, by 4; required the numbers. Ans. 6 and 8.

25. What two numbers are those, which being both multiplied by 27, the first product is a square, and the second the root of that square; but being both multiplied by 3, the first product is a cube, and the second the root of that cube? Ans. 243 and 3.

26. A man bought a horse, which he sold, after some time, for 24 dollars. At this sale he loses as much per cent. upon the price of his purchase as the horse cost him. What did he pay for the horse?

Ans. He paid $60 or $40; the problem does not decide which sum.

27. What two numbers are those whose product is equal to the difference of their squares; and the greater number is to the less as 3 to 2? Ans. No such numbers exist. 28. What two numbers are those, the double of whose product is less than the sum of their squares by 9, and half their product is less than the sum of their squares by 9?

Ans. The numbers are 9 and 12. Will the student show that examples 24 and 28 are essentially the same.

SECTION V.

ARITHMETICAL PROGRESSION.

(ART. 105.) A SERIES of numbers or quantities, increasing or decreasing by the same difference, from term to term, is called arithmetical progression.

Thus, 2, 4, 6, 8, 10, 12, &c., is an increasing or ascending arithmetical series, having a common difference of 2; and 20, 17, 14, 11, 8, &c., is a decreasing series, whose common difference is 3.

We can more readily investigate the properties of an arithmetical series from literal than from numeral terms. Thus, let a represent the first term of a series, and d the common difference. Then

a, (a+d), (a+2d), (a+3d), (a+4d), &c., represents an ascending series; and

a, (a—d), (a—2d), (a—3d), (a—4d), &c.,

represents a descending series.

Observe that the coefficient of d in any term, is equal to the number of the preceding term.

The first term exists without the common difference. All other terms consists of the first term and the common difference multiplied by one less than the number of terms from the first.

Thus, if the first term of an arithmetical series is a, and d the common difference, the tenth term would be expressed by

When the series is decreasing, the sign to the term containing d will be minus, the 20th term, for example, would be

We add a few examples to exercise the pupil in finding any term of a series, when the first term, a, and the common difference, d, are given.

1. When a 2 and d=3, what is the 10th term?

Ans. 30.

2. When a 3 and d=2, what is the 12th term?

3. When a 7 and d=10, what is the 21st

Ans. 25.

term?

Ans. 207.

4. When a=1 and d=1, what is the 100th term?

Ans. 50.

5. When a =3 and d=1, what is the 100th term ?

Ans.

6. When a=0 and d=1, what is the 89th term?

36.

Ans. 11.

7. When a= 6 and d=-1, what is the 20th term?

Ans.-31.

8. When a 30 and d=-3, what is the 31st term?

Ans. -60.

Wherever the series is supposed to terminate, is the last term, and if such term be designated by L, and the number of terms by n, the last term must be a+(n—1)d, or

a-(n-1)d, according as the series may be ascending or descending, which we draw from inspection.

Hence, .

L=a±(n-1)d

(A)

(ART. 106.) It is manifest, that the sum of the terms will be the same, in whatever order they are written.

Take, for instance, the series 3, 5, 7, 9, 11,

11, 9, 7, 5, 3.

The sums of the terms will be 14, 14, 14, 14, 14.

ad, a+2d, a+3d, a+4d,

Take

. a

Inverted,

Sums,

2a+4d, 2a+4d, 2a+4d, 2a+4d, 2a+4d.

Here we discover the important property, that, in arithmetical progression, the sum of the extremes is equal to the sum of any other two terms equally distant from the extremes. Also, that twice the sum of any series is equal to the extremes, or first and last term repeated as many times as the series contains terms.

Hence, if S represents the sum of a series, and n the number of terms, a the first term, and L the last term, we shall have 2S=n(a+L) S=12 (a+L)

Or,

(B)

The two equations (A) and (B) contain five quantities, a, d, L, n, and S; any three of them being given, the other two can be determined.

Two independent equations are sufficient to determine two unknown quantities (Art. 45), and it is immaterial which two are unknown, if the other three are given.

By examining the two equations they will become familiar.