16. Find two numbers, such that their sum, their product, and the difference of their squares, may all be equal to one another. Ans.V5, and 2 ± √5.

17. Divide the number 100 into two such parts, that the sum of their square roots may be 14. Ans. 64 and 36.

18. Divide the number 40 into two such parts, that twice their product shall be equal to the cube of the less number. Ans. 8 and 32.

19. A and B hired a pasture, into which A put 4 horses, and B as many as cost him 18 shillings a week. Afterwards B put in two additional horses,. and found that he must pay 20 shillings a week; at what rate was the pasture hired?

Ans. B had 6 horses in the pasture at first, and the price of the whole pasture was 50 shillings per week.

20. A mercer bought a piece of silk for £16. 4s., and the number of shillings he paid per yard, was to the number of yards as 4 to 9. How many yards did he buy, and what was the price per yard?

Ans. 27 yards at 12 shillings per yard. 21. There are two numbers in the proportion of 4 to 5, and the difference of their squares is 81. What are the numbers? Ans. 12 and 15.

In

22. A person bought for a dollar, as many pounds of sugar as were equal to half the number of dollars he laid out. selling the sugar, he received for every 100lbs. as many dollars as the whole had cost him, and he received on the whole $20,48. How many dollars did he lay out, and what did he give for a pound?

Ans. He laid out $16, and gave 12 cents per pound. 23. What two numbers are those, whose difference multiplied by the greater, produces 40, and by the less, 15?

Ans. 8 and 3.

24. 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. 25. Find two numbers, such that the less may be to the greater as the greater is to 12, and the sum of their squares Ans. 3 and 6.

45.

26. What two numbers are those whose product is 20, and the difference of their cubes 61? Ans. 4 and 5.

27. What number is that, which being divided by the product of its two digits, the quotient is 5; and 9 being subtracted from the number itself, the digits will be inverted? Ans. 32.

23. Divide 20 into three such parts, that the continual product of these parts may be 270; and that the difference of the first and second may be 2 less than the difference of the second and third. Ans. 5, 6, and 9.

29. Divide the number 11 into two product of their squares may be 784.

such parts, that the Ans. 4 and 7.

30. Find two numbers, such that the sum of their squares may be 89, and their sum multiplied by the greater may produce 104. Ans. 5 and 8.

CHAPTER V.

PROGRESSIONS AND LOGARITHMS.

ARITHMETICAL PROGRESSION.

(78.) An arithmetical progression, or progression by differences, is a series of numbers, in which the terms continually increase or decrease by a constant number, which is called the ratio or common difference of the progression, (Arith. 60).

Thus, 1, 3, 5, 7, 9, 11, &c. is an increasing progression ; and, 11, 9, 7, 5, 3, 1, is a decreasing progression. The ratio or common difference in these progressions is 2. In order to investigate the properties of arithmetical progressions, we will take the general progression,

the first term of which is a, the last term 7, the common difference q, and the number of terms n.

(79.) From the nature of an arithmetical progression, we have,

b=a+q, c = b + q, d = c + q, e=d+q, &c. ; or, by putting the value of b in the equation c= b+q and the value of c as found in this, in the next, and so on, we shall have,

b = a + q, c = a+2q, d = a + 3q, e = a + 4q, &c. ; and writing these values in the general progression, it becomes,

a, a + q, a +2q, a +3q, a +4q, &c.,

from which it will be seen, that any term of the progression may be found by adding to a as many times the common difference as there are terms before this term, or as many times the common difference as there are terms in the progression less ONE. Hence, to find the last term of a progression, we have the formula,

If the progression be a decreasing one, we have only to make q negative, and the formula will become,

૧

From these formulas is derived the principle laid down in arithmetic (62), namely:

In an increasing arithmetical progression, the last term is equal to the first term, plus the common difference multiplied by the number of terms less 1; and in a decreasing progression the last term is equal to the first, minus the common difference multiplied by the number of terms less 1.

(80.) Let' denote any term of a progression which has p terms before it, and y a term which has p terms after it, then, by the last formula, we shall have,

And adding these equations together, we obtain the equation,

x + y = a + 1,

which shows that the sum of any two terms, equally distant from the extremes, is equal to the sum of the extremes (Arith. 61).

(81.) If x be the middle term of a progression, having an odd number of terms, then

and adding, we obtain 2x=a+1.

In this case, therefore, the sum of the extremes is equal to double the middle term (Arith. 61).

(82.) If, from the equation = a +(n-1)2, we find the value of 9, we shall have,

or, the common difference of an arithmetical progression, equal to the difference of the extremes divided by the number of terms less 1.

(83.) In order to find the sum of any number of terms of an arithmetical progression, let S= the sum, and we shall have, a+b+c+d+e+f... +i+k+ l,

S=

Since the parts (a + b) (b + k) &c. are equal to each other, each being the sum of two terms equally distant from the extremes, and therefore equal to (a+1), the sum of the extremes (80), we shall have in a progression of n terms,

Hence we conclude, that the sum of the terms of an arithmetical progression is equal to the sum of the extremes, multiplied by the number of terms, and divided by 2 (Arith. 64).

other expressions may be derived; we will give some of the most useful.

(85.) From the first we have, by considering all the quantities which enter into it, except n, known,

that is, the number of terms equal to the difference of the extremes divided by the common difference, plus 1.

(86.) From the same equation, we obtain, by transposing a and 7, and changing the signs,

or, the first term equal to the last term, minus the common difference multiplied by the number of terms less 1.

(87.) The equation S

=

(a + 1)n
2

may be changed so as

to express each of the quantities S, a, l, and n, in terms of the others. Thus, by making n, l, and a, successively, to stand alone in one member of the equation, we obtain the three following equations :

From the first of these we conclude, that if the sum of a progression and the extremes are known, we shall find the number of terms by dividing twice the sum by the sum of the extremes.

From the second and third, that the sum, the number of terms, and one extreme being given, the other extreme may be found by dividing twice the sum by the number of terms, and subtracting the given extreme from the quotient.

(88.) If in the equation S=

(a-1)n
2

we substitute in the

place of l, its value, as found in the equation = a (= a +(n+1)q,

a formula, from which we can find the sum of a progression, when the first term, the number of terms, and the common difference, are known.