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As m may

Cor. I.



number integral or fractional, positive or negative; hence the division, involution and evolution of a binomial, may be performed by this theorem.



Ex. I.

Let m=;, then a+5)* =a*+

+-Xa #+, &c. This being applied to the extra&tion of the square root of a2 + x2 (by inserting afor a and *? for b) the same series results as formerly, (Chap. IV.).



Ex. 2.

If %, is to be turned into an infinite series, fince ==1X1—?, let a=1, b=-r, and m=-1; and the same a b

I series will arise as was obtained by division. (Chap. I.).


212=zz=rax 2r2—241 may be expressed by an infinite series, supposing a=2rz, b=, and m=-, and then multiplying that series by ra.

In like manner,

manner, v zrz-z?


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I, +

2 XI

IXI 2.2 * 1.2

Cor. ż. This theorem is useful also in discovering the law of an infinite series, produced by division or evolution. Thus, the series expressing the square root of a2 + x?, consists of a, together with a series of fractions, in the numerators of which are the even powers of x, and in the denominators the odd powers of a.

The numeral coefficients of the terms of the whole series, as deduced by the theorem, will be:

I X1.3

2.2.2 X 1.2.3 IX 1.3:5

&c. the point being used (as X' it often is) to express the product of the numbers between which it is placed. The law of continuation is obvious, and the feries

may be carried on to any number of terms, without using the theorem. Herice also the coefficient of the nth term is IX 1.3.5 &c. .. (n-2 terms)

; and it is +, if n X 1. 2. 3. 4. &c. (n-1) is an even number, and -, if n is odd.

Note. If the binomial is d+b, the signs of the terms of any power are all positive: If it is a--b, the alternate terins are negaS


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tive, beginning at the second. This theorem may be applied to quantities which consist of more than two parts, by fupposing them distinguished into two, and then substituting for the powers of these com-pound parts, their values, to be obtained also, if required, from the theorem. Thus, a+b+c=a+b+c2


An infinite series may itself be multiplied or divided by another; it may

be involved or evolved, and various other operations may be performed upon it which are necessary in the higher parts of algebra. The methods for finding the sum depend upon other principles.

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IV. Properties of Numbers.

Theor. I. The sum of two quantities multiplied by their difference is equal to the difference of their squares.


Let the quantities be represented by a and b, then a +bxa-b=a_b?, as appears by performing the operation.

Cor. If a and b be any two quantities of which the sum may be denoted by s, the difference by d, and their product by P, then the following propositions will be true.

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1. a? +b?=s?-2p. 2. a?-basd.
3. a3 +63 = 33-3PS. 4. a3.63=59ddp.
5. a4+b4=54-4p2 +282 6. 4--64553d2sdp, &c.


It is unnecessary to express these propofitions in words, and the demonstrations are very easy, by raising atb to certain powers, and making proper substitutions.

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THEOR. II. The sum of


number of terms (n) of the odd numbers 1, 3, 5, &c. beginning with 1, is equal to the square of that number (n).

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In the rule for summing an arithmetical series, let a =1,b=2, and n En, and the sum



of this series will be s

2an+n?b--nb 2n2


Q. E. D.
THEOR. III. The difference of

any two
square numbers is equal to the sum of the
two roots, together with twice the sum of
the numbers in the natural scale between
the two roots.

Let the one number be p, and the other
p+n, the intermediate numbers are p+1,
p+2, ... &c. p+n-1. The difference

. ptn
of the squares of the given numbers is
2pn+1?; the sum of the two 'roots is 2p+n,
and twice the sum of the series pti+p+2

&c. P+1-I is (by Cor. 1. ist fect. of
this chap.) 25=2p+nXn-I, viz. the sum
of the first and last multiplied by the num-
ber of terms, and it is plain that 2punt
2p+nxn--I=2pntn". Therefore, &c.

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Let r. be any number, and n


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any integer, r--i is divisible by rấI.
The quotient will be r

&c. till
the index of r be o, and then the last term



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