CHAPTER VII.

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

2d. 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 to each other, with respect to their difference; and GEOMETRICAL PROPORTION, the relation of quantities to each other, with respect to their quotient.

Of Arithmetical Proportion.

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. The

first and fourth terms are called the extremes; and the second and third terms, the means.

Let a, b, c, and d, denote four quantities in arithmetical proportion; and d the difference between either antecedent and its consequent.

Then,

also,

abd, and a = b+d;

cdd, and dcd.

By adding the last two equations, we have

a+d=b+c: that is,

If four quantities are in arithmetical proportion, the sum of the two extremes is equal to the sum of the two means.

161. When the difference between the first antecedent and consequent is the same as between any two consecutive terms of the proportion, the proportion is called an arithmetical progression. Hence, an arithmetical progression, or a progression by differences, is a succession of terms, each of which is greater or less than the one that precedes it by a constant quantity, which is called the common difference of the progression. Thus,

and

1, 4, 7, 10, 13, 16, 19, 22, 25,

60, 56, 52, 48, 44, 40, 36, 32, 28,

are arithmetical progressions. 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. An arithmetical progression, is also called, an arithmetical series; and generally,

A series is a succession of terms derived from each other according to some fixed and known law.

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

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 d represent the common difference of the progression e. f. g. h. k, &c.,

a. b C.

which we will consider increasing.

From the definition of a progression, it follows that,

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

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 be any term, and n the number which marks the place of it. Then, the number of preceding terms will be deno1, and the expression for this general term, will be

ted by n

That is, any term is equal to the first term, plus the product of the common difference by the number of preceding terms.

If we make n = 1, we have la; that is, the series will have but one term.

If we make

n = 2, we have l=a+d;

that is, the series will have two terms, and the second term is equal to the first plus the common difference.

serves to find any term whatever, without determining all those which precede it.

Thus, to find the 50th term of the progression,

163. If the progression were a decreasing one, we should have la (n - 1)d.

=

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

EXAMPLES.

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

− l la (n-1)d gives = 60 (201) 360-573.

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2. The first term is 90, the common difference 4: what is the 15th term? Ans. 34.

3. The first term is 100, and the common difference 2: what is the 40th 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. i. k., be the proposed progression,

Let a.

b

C.

e.f

and n the number of terms.

We will first observe that, if x denote a term which has p terms before it, reckoning from the first term, and y a term which has p terms before it, reckoning from the last term, we have, from what has been said,

Now, to find the sum of all the terms, write the progression below itself, but in an inverse order, viz.,

Calling S the sum of the terms of the first progression, 2S will be the sum of the terms in both progressions, and we shall have

2S=(a+1)+(b + k) + (c + i)... + (i + c) + (k + b) + (l + a). And, since all the parts a, b + k, c + i

are equal to each other, and their number equal to n, by which we designate the number of terms in each series, we have

That is, the sum of the terms of an arithmetical progression, is equal to half the sum of the two extremes multiplied by the number of terms.

EXAMPLES.

1. The extremes are 2 and 16, and the number of terms 8: what is the sum of the series?

2. The extremes are 3 and 27, and the number of terms 12: what is the sum of the series?

Ans. 180.

3. The extremes are 4 and 20, and the number of terms 10: what is the sum of the series?

Ans. 120.

4. The extremes are 8 and 80, and the number of terms 10: what is the sum of the series?

Ans. 440.

contain five quantities, a, d, n, 1, and S, and consequently give rise to the following general problem, viz.: Any three of these five quantities being given, to determine the other two.