456. Now let n=1, =1, (a+x);=(a+x)2=V/(a+x);

and aa; hence the series (Art. 454) is trans

formed into (a+x)=(1+1)+()

+ (1—r).(1 —— 2r) (27) +&c.)

2.3.r3

Let a=1, b=1; then /2=1+;+

+ (1 —r). (1—2r) + &c.

+

2.3.
p3

+&c.

+&c. And if r=3,

1 1 5 2.5 2.11

(B).

2.7.11

By means of the series marked A, the rth root of many other numbers may be found; if a and ≈ be so assumed, that x is a small number with respect to a, and a, a whole number.

Ex. 3. It is required to convert 5, or its equal (4+1), into an infinite series.

Here a=4, x=1, r=2; then a√4=2, and we

1

have √(4+1)=2(1+23—27+210

Ex. 4. It is required to convert 9, or its equal (8+1), into an infinite series.

Here a=8, x=1, r=3; then Va=38=2, and

we obtain (8+1)=2/9=2} 1 +

1

3.8 32.82

457. The several terms of these series are found by substituting for a, x, and r, their values in the general series marked (A) or (B), and then rejecting the factors common to both the numerators and denominators of the fractions.

Thus, for instance, to find the 5th term of the series expressing the approximate value of 9, we take the 5th term of the general series marked (A), ̧(1 --r).(1—2r).(1—3r) where a=8,

which is

3 3 4x4

x=1, and r=3; .. the value of the fraction is

2.5.8

2.3.4.34

2 .5 35.84

In this manner each term of the series

is calculated; and the law which they observe is, that the numerators of the fractions consist of certain combinations of prime numbers, and the denominators of combinations of certain powers of a and r.

Ex. 5. Find the value of (ca-x)in a series.

5x6

-&c.)

22. c2 25.c 27.c®

Ex. 6. It is required to convert 2/6, or its equal (8-2), into an infinite series.

Ex. 7. It is required to extract the square root of 10, in an infinite series.

1.3

2.4.6.3

Ex. 8. To expand a2 (a2 - x) in a series.

+ &c.

3.5

Ex. 9. To find the value of {/(α3 +x3) in a

Ex. 10. Find the cube root of 1-x3, in a series.

Ans. 1

X3 x 5x9 10x12

-

3

9 81 243

&c.

CHAPTER XIII.

ON

PROPORTION AND PROGRESSION.

§ I. ARITHMETICAL PROPORTION AND PROGRESSION.

458. ARITHMETICAL PROPORTION is the relation which two numbers, or quantities, of the same kind, have to two others, when the difference of the first pair is equal to that of the second.

459. Hence, three quantities are in arithmetical proportion. when the difference of the first and second is equal to the difference of the second and third. Thus, 2, 4, 6; and a, a+b, a+2b, are quantities in arithmetical proportion.

460. And four quantities are in arithmetical proportion, when the difference of the first and second is equal to the difference of the third and fourth. Thus, 3, 7, 12, 16; and a, a+b, c, c+b, are quantities in arithmetical proportion.

461. ARITHMETICAL PROGRESSION is, when a series of numbers or quantities increase or decrease by the same common difference. Thus 1, 3, 5, 7, 9, &c. and a, a+d, a+2d, a+3d, &c. are an increasing series in arithmetical progression, the common differences of which are 2 and d. And 15, 12, 9, 6, &c. and a, a-d, a-2d, a-3d, &c. are decreasing series in arithmetical progression, the cominon differences of which are 3 and d.

462. It may be observed, that GARNIER, and other European writers on Algebra, at present, treat of arithmetical proportion and progression under the denomination of equi-differences, which they consider, as BONNYCASTLE justly observes, not without reason, as a more appropriate appellation than the former, as the term arithmetical conveys no idea of the nature of the subject to which it is applied.

463. They also represent the relations of these quantities under the form of an equation, instead of by points, as is usually done; so that if a, b, c, d, taken in the order in which they stand, be four quantities in arithmetical proportion, this relation will be expressed by a-b-c-d; where it is evident that all the properties of this kind of proportion can be obtained by the mere transposition of the terms of the equation.

464. Thus, by transposition, a+d=b+c. From which it appears, that the sum of the two extremes is equal to the sum of the two means: And if the third term in this case be the same as the second, or c=b, the equi-difference is said to be continued, and we have

a+d=2b; or b= 1⁄2 (a+d) ;

where it is evident, that the sum of the extremes is double the mean; or the mean equal to half the sum of the extremes.

465. In like manner, by transposing all the terms of the original equation, a-b-c-d, we shall have b-a-d-c; which shows that the consequents b, d

can be put in the places of the antecedents a, c; or, conversely, a and c in the places of b and d.

466. Also, from the same equality a-b-c-do there will arise, by adding m-n to each of its sides, (a+m)−(b+n)=(c+m)—(d+n);

where it appears that the proportion is not altered, by augmenting the antecedents a and by the same quantity m, and the consequents bed by another quantity n. In short, every operation by way of addition, subtraction, multiplication, and division, made upon each member of the equation, a-b=c -d, gives a new property of this kind of proportion, without changing its nature.

467. The same principles are also equally applicable to any continued set of equi-differences of the forin a-bb-c=c-d=d—e, &c. which denote the relations of a series of terms in what has been usually called arithmetical progression.

468. But these relations will be more commodiously shown, by taking a, b, c, d, &c. so that each of them shall be greater or less than that which precedes it by some quantity d'; in which case the terms of the series will become

a, a±d, a±2d', a±3d', a±4d, &c.

Where, if be put for that term in the progression of which the rank is n, its value, according to the law here pointed out, will evidently be

l=a±(n−1)ď';

which expression is usually called the general term of the series; because, if 1, 2, 3, 4, &c. be successively substituted for n, the results will give the rest of the terms.

Hence the last term of any arithmetical series is equal to the first term plus or minus, the product of the common difference, by the number of terms less one.

469. Also, ifs be put equal to the sum of any number of terms of this progression, we shall have x+(a±ď)+(a±2ď) + .... +[a±(n−1)d'].