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Cor. I. As m may
denote any number integral or fractional, pofitive or negative; hence the divifion, involution and evolution of a binomial, may be performed by this theorem.
Ex. I. Let m=1⁄2, then a+b|2=a3+ ÷a ̄ ̄ ̄1b+÷×—÷×a ̄3⁄4b2+, &c. This
being applied to the extraction of the fquare root of a2+x2 (by inserting a2 for a and x2 for b) the fame feries refults as formerly. (Chap. IV.).
If is to be turned into an
infinite series, since_=1×1, let
a=1, b——r, and m——1; and the fame feries will arise as was obtained by division. (Chap. I.).
In like manner,
'2rz—z2=r2 × 2rx—x2 + may be expreffed by an infinite feries, fuppofing a=2rz, b=-x2, and m= then multiplying that series by r2.
Cor. 2. This theorem is ufeful alfo in discovering the law of an infinite feries, produced by divifion or evolution. Thus, the feries expreffing the fquare root of a2+ x2, confifts 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 nume
ral coefficients of the terms of the whole feries, as deduced by the theorem, will be:
188.8.131.52 × 184.108.40.206.'
&c. the point being used (as
it often is) to exprefs 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 ufing the theorem. Hence
alfo the coefficient of the nth term is I×1.3.5 &c.
X I. 2. 3.4 &c. (1)
; and it is, if n
is an even number, and -, if n is odd.
Note. If the binomial is a+b, the figns of the terms of any power are all positive: If it is ab, the alternate terms are negative,
tive, beginning at the fecond. This theorem may be applied to quantities which 'confift of more than two parts, by fuppofing them diftinguished into two, and then fubftituting for the powers of these com-pound parts, their values, to be obtained alfo, if required, from the theorem. Thus, a+b+c2=a+b+c2.
An infinite feries 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 neceffary in the higher parts of algebra. The methods for finding the fum depend upon other principles.
IV. Properties of Numbers.
THEOR. I. The fum of two quantities multiplied by their difference is equal to the difference of their fquares.
Let the quantities be reprefented by a and b, then a+bxa—b±a2—b2, as appears by performing the operation.
COR. If a and b be any two quantities of which the fum may be denoted by s, the difference by d, and their product by p, then the following propofitions will be true.
5. a4+b4=s4—4ps2+2p2 6. a4 b4 s3d-2sdp, &c.
It is unneceffary to exprefs these propofitions in words, and the demonftrations are very easy, by raising a+b to certain powers, and making proper fubftitutions.
THEOR. II. The fum of any number of terms (2) of the odd numbers 1, 3, 5, &c. beginning with 1, is equal to the fquare of that number (n).
In the rule for fumming an arithmetical series, let a = 1, b2, and nn, and the sum
of this feries will be s
Q. E. D.
THEOR. III. The difference of any two fquare numbers is equal to the fum of the two roots, together with twice the fum of the numbers in the natural fcale 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 of the fquares of the given numbers is 2pn+n; the fum of the two roots is 2p+n, and twice the fum of the feries p++p+?
&c. p+n-1 is (by Cor. 1. 1ft fect. of this chap.) 25=2p+n×n−1, viz. the fum of the first and laft multiplied by the number of terms, and it is plain that 2p+n+ 2p+nxn−1=2pn+n. Therefore, &c.
LEM. I. Let be any number, and n
any integer, r—I is divisible by r—1.
The quotient will be r
the index of r be o, and then the last term