:

if a b c d; that is, the roots of the same degree of four proportional quantities, are also proportional.

Such are the leading principles in the theory of proportion. This theory was invented for the purpose of discovering certain quantities by comparing them with others.. Latin names were for a long time used to express the different changes or transformations, which a proportion admits of. We are beginning to relieve the memory of the mathematical student from so unnecessary a burden; and this parade of proportions might be entirely superseded by substituting the corresponding equations, which would give greater uniformity to our methods, and more precision to our ideas.

228. We pass from proportion to progression by an easy transition. After we have acquired the notion of three quantities in continued equidifference, the last of which exceeds the second, as much as this exceeds the first, we shall be able, without difficulty, to represent to ourselves an indefinite number of quantities, a, b, c, d, &c., such, that each shall exceed the preceding one, by the same quantity d, so that

b = a + d, c = b + &, d = c + &, e = d + ô, &c. A series of these quantities is written thus ;

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

and is termed an arithmetical progression; I have thought it proper, however, to change this denomination to that of progression by differences. (See Arith. art. 127, note.)

We may determine any term whatever of this progression, without employing the intermediate ones. In fact, if we substitute for b its value in the expression for c, we have

c = a + 28;

by means of this last, we find

da 36, then e = a + 48,

and so on; whence it is evident, that representing by 7 the term, the place of which is denoted by n, we have

Let there be, for example, the progression

3.5.7.9 11 13. 15. 17, &c.

here the first term a 3, the difference or ratio d = 2; we find for the eighth term

3 (81) 2 = 17,

the same result, to which we arrive by calculating the several preceding terms.

The progression we have been considering is called increasing ; by reversing the order, in which the terms are written, thus, 17. 15. 13. 11 9.7 5 3 1.

1.

3, &c.

we form a decreasing progression. We may still find any term whatever by means of the formula a + (n-1) d, observing only, that d is to be considered as negative, since, in this case, we must subtract the difference from any particular term in order to obtain the following.

229. We may also, by a very simple process, determine the sum of any number whatever of terms in a progression by differences. This progression being represented by

and S denoting the sum of all the terms, we have

S = a+b+c.

+i+k+l.

Reversing the order, in which the terms of the second member of this equation are written, we have still

If we add together these equations, and unite the corresponding terms, we obtain

2S=(a+1)+(b+k)+(c+i)....+(i+c)+(k+b)+(l+a);

but by the nature of the progression, we have, beginning with the first term,

a + 8 = b, b + d = c, . . . . . i+d=k, k + 8 = 1, and, consequently, beginning with the last

1—8—k, k—di,..... c-8b, b-da; by adding the corresponding equations, we shall perceive at once, that

furnishes us with the means of finding any two of the five quantities, a, d, n, l, and S, when the other three are known; I shall not stop to treat of the several cases, which may be presented.

231. From proportion is derived progression by quotients or geometrical progression, which consists of a series of terms, such, that the quotient arising from the division of one term by that which precedes it, is the same, from whatever part of the series the two terms are taken. The series

2: 6:18: 54: 162: &c.

45:15: 5: 5: : &c.

are progressions of this kind; the quotient or

ratio is 3 in the first, and in the other; the first is increasing, and the second decreasing. Each of these progressions forms a series of equal ratios, and for this reason is written, as above.

be the terms of a progression by quotients; making

have by the nature of the progression,

a

=q, we

or b = aq,

c = b 9, d = cq, e = dq,... 1

=

kq.

Substituting, successively, the value of b in the expression for c, and the value of c in the expression for d, &c., we have

b = aq, c = a q3, d = a q3, e = a q1, . . . l = a q′′−1, taking n to represent the place of the term l, or the number of terms considered in the proposed progression.

By means of the formula 1 = a q-1 we may determine any term whatever, without making use of the several intermediate ones. The tenth term of the progression

2:6:18: &c.,

for example, is equal to 2 × 3o = 39366,

232. We may also find the sum of any number of terms we please of the progression

abcd, &c.

by adding together the equations

b = aq, c = bq, d = c q, e = d q,

for the result will be

...

. l = kq ;

b+c+d+e +1 = (a+b+c+d. + k) q;

and representing by S the sum sought, we have

...

b+c+d+e.... + 1 = Sa,
a+b+c+d....+k=S-1,
Saq (S-1),

whence

and, consequently,

S=

+ The truth of this result may be rendered very evident, independently of analysis. If it were required, for example, to find the sum of the progression

26:18: 54: 162,

multiplying by the ratio, we have

6:18:54: 162: 486.

The first series being subtracted from this gives 486-2, equal to so many times the first series, as is denoted by the ratio minus one, that is

If we multiply by the ratio q the general series

we have

abcd: e..... l,

:: a q : b զ : c q: dq: eq ..... 1 q.

1

2

Then, because b = aq, &c., the second series minus the first is lq — α, equal to so many times the first series, as is denoted by the ratio

In the above example, we find for the sum of the first ten terms of the progression

comprehend the mutual relations, which exist among the five quantities, a, q, n, 1, and S, in a progression by quotients, and enable us to find any two of these quantities, when the other three are given.

234. If we substitute a q-1 in the place of 1, in the expression for S, we have

When q is a whole number, the quantity q" will become greater and greater in proportion to the increased magnitude of the number n; and S may be made to exceed any quantity whatever, by assigning a proper value to n, that is, by taking a sufficient number of terms in the proposed progression. But if q is a fraction, represented by, we have

1

m

and it is evident, that as the number n becomes greater, the term

mn-1

will become smaller, and, consequently, the value of S will

therefore, the greater the number of terms we take in the proposed progression, the more nearly will their sum approach to

+ Multiplying the numerator and denominator by -m.