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QUESTIONS.

1. There are two numbers whose difference is 15, and half their product is equal to the cube of the lesser number. What are those

2. What two numbers are those whose sum, multiplied by the greater, is equal to 77; and whose difference, multiplied by the lesser, is equal to 12?

Ans. 4 and 7, or a ^2 and u y/2.

3. To divide 100 into two such parts, that the sum of their square roots may be 14. Ans. 64 and 36.

4. It is required to divide the number 24 into two such parts, that their product may be equal to 35 times their difference.

Ans. 10 and 14.

5. The sum of two numbers is 8, and the sum of'their cubes is 152. What are the numbers? Ans. 3 and 5.

6. The sum of two numbers is 7, and the sum of their 4th powers is 641. What are the numbers? Ans. 2 and 5.

7. The sum of two numbers is 6, and the sum of their 5th powers is 1056. What are the numbers? Ans. 2 and 4.

8. Two merchants each sold the same kind of stuff; the second sold 3 yards more of it than the first, and together, they receive 35

numbers?

Ans. 3 and 18.

crowns. The first said to the second, I would have received 24

crowns for your stuff; the other replied, and I would have received

12J crowns for yours. How many yards did each of them sell?

( 1st merchant x=15 x=5 i

Ans. { , or >

I 2d . . . y=18 y=8. )

9. A widow possessed 13,000 dollars, which she divided into two parts, and placed them at interest, in such a manner, that the incomes from them were equal. If she had put out the first portion at the same rate as the second, she would have drawn for this part 360 dollars interest; and if she had placed the second out at the same rate as the first, she would have drawn for it 490 dollars interest. What were' the two rates of interest?

Ans. 7 and 6 per cent.

CHAPTER IV. N

Of Proportions and Progressions.

159. Two quantities of the same kind may be compared together in two ways :—

1st. By considering how much one is greater or less than the other, which is shown by their difference; and

2ndly. By considering how many times one is greater or less than the other, which is shown by their quotient.

Thus, in comparing the numbers 3 and 12 together with respect to their difference, we find that 12 exceeds 3 by 9; and in comparing them together with respect to their quotient, we find that 12 contains 3 four times, or that 12 is 4 times as great as 3.

The first of these methods of comparison is called Arithmetical Proportion, and the second, Geometrical Proportion. Hence,

Arithmetical Proportion considers the relation of quantities with respect to their difference, and Geometrical Proportion the relation of quantities with respect to their quotient.

Of Arithmetical Proportion and Progression.

160. If we have four numbers, 2, 4, 8, and 10, of which the difference between the first and second is equal to the difference between the third and fourth, these numbers are said to be in arithmetical proportion. The first term 2 is called an antecedent, and the second term 4, with which it is compared, a consequent. The number 8 is also called an antecedent, and the number 10, with which it is compared, a consequent.

161. When the difference between the first antecedent and consequent is the same as between any two adjacent terms of the proportion, the proportion is called an arithmetical progression. Hence, a progression by differences, or an arithmetical progression, is a series in which the successive terms continually increase or decrease by a constant quantity, which is called the common difference of the progression.

Thus, in the two series

1, 4, 7, 10, 13, 16, 19, 22, 25, . . . 60, 56, 52, 48, 44, 40, 36, 32, 28, . . .

the first is called an increasing progression, of which the common difference is 3, and the second a decreasing progression, of which the common difference is 4.

In general, let a, b, c, d, e,f, . . . designate the terms of a progression by differences; it has been agreed to write them thus:

a.b.c.d.e.f.g.h.i.k . . .

This series is read, a is to b, as b is to c, as c is to d, as d is to e, &c. This is a series of continued equi-differences, in which each term is at the same time a consequent and antecedent, with the exception of the first term, which is only an antecedent, and the last, which is only a consequent.

162. Let r represent the common difference of the progression

a.b.c.d.e.f.g.h, &c,

which we will consider increasing.

From the definition of the progression, it evidently follows that

b=a+r, c=b+r=a+2r, d=c+r=a+3r;

and in general, any term of the series, is equal to the first term plus as many times the common difference as there are preceding terms.

Thus, let 1 be any term, and n the number which marks the place of it, the expression for this general term, is

l = a+(n-l)r. .

That is, the last term is equal to the first term, plus the product of the common difference by the number of terms less one.

If we make »=1, we have I—a; that is, the series will have but one term.

If we make n=2, we have l~a-\-r; that is, the series will have two terms, and the second term is equal to the first plus the common difference.

EXAMPLES.

1. If 0=3 and r=2, what is the 3rd term? Atts. 7.

2. If a=5 and r=4, what is the 6th term? Ans. 25.

3. If a=7 and r=5, what is the 9th term? Ans. 47.

The formula l—a-\-(n— l)r, serves to find any term whatever, without our being obliged to determine all those which precede it.

Thus, by making re=50, we find the 50th term of the progression, 1 . 4 . 7 . 10 . 13 . 16 . 19 . . . . in which 7=1+49x3=148. The 60th term of the progression,

1 . 5 . 9 . 13 . 17 . 21 . 25 . . . . gives Z=l+59x4=237.

163. If the progression were a decreasing one, we should have

l—a~ (re—l)r.

That is, in a decreasing arithmetical progression, the last term is equal to the first term minus the product of the common difference by the number of terms less one.

EXAMPLES.

1. The first term of a decreasing progression is 60, the number of terms 20, and the common difference 3: what is the last term?

l=a-(n-l)r gives Z=60—(20-1)3 = 60-57=3.

2. The first term is 90, the common difference 4, and the number of terms 15: what is the last term? Ans. 34.

3. The first term is 100, the number of terms 40, and the com' mon difference 2: what is the last term? Ans. 22.

164. A progression by differences being given, it is proposed to prove that, the sum of any two terms, taken at equal distances from the two extremes, is equal to the sum of the two extremes.

Let a.b.c.d.e.f. . . . i. k . I he the proposed progression, and n the number of terms.

We will first observe that, if x denotes a term which has^t terms before it, and y a term which has p terms after it, we have, from what has been said, x=a-\-pXr, and y=l —pXr;

whence, by addition, x+y=.a-\-l, which demonstrates the proposition.

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