Q=(ab+ac+ad+ae+be+bd+&c.); the fourth Rx-3, where R=(abc+abd+acd+ace+bcd+bce+&c.); the mth term Sx"―+1, where S=sum of the combinations of any m~ 1 quantities; &c. &c.; and the product itself would be expressed by "+Px+Qx2+ Rx-3.....+Sx"+",

n(n-1)

146. Now P consists of the sum of all the quantities a, b, c, d, e, &c. the number of which is n; Q of the sum of the combinations of any two of those quantities, the number of which (by Art. 143) is ; R of the sum of the combinations of any three of those quantities, the number of which n(n-1)(n-2)

is

1.2.3

2

; and S of the sum of the combinations of

any (m-1) of those quantities, the number of which is

147. Let a=b=c=d=e, &c. then (x+a)(x+b) (x+c) (x+d), &c. becomes (x+a)", P becomes na, becomes n(n-1)a'

n(n-1) (n-2)

; R becomes

a, and S (the coeffi

2

1.2.3

cient of the mth term) becomes

n(n−1)(n−2)... (n−m+2)

∙am

1.2.3...m-1

148. From hence it appears that (x+a)" (x" + Px"−1

+Qx"¬+Rx"~3

=

+Sx+) x2+nax"'+

and (preserving the notation adopted in Chap. III.) we

149. From the inspection of this series it is evident, that if n be a positive whole number, it will terminate after (n+1) terms; for let m=n+2, then n−m+2=0, and consequently the coefficient which involves the factor (n-m+2) vanishes. Let m=n+1, then n-m+2=1,n−m+1=0, and m — 1 n(n-1)(n-2)....3.2.1

=n;.. the (n+1th) (or last) term is 1.2.3 ....

(n-1) n ab" or b". If n be fractional or negative, the series will not terminate, and in this case the value of any expanded binomial can only be expressed in the form of an infinite series.

150. If in the series expressing the value of (a+b)", for b we put-b, then those terms which involve the odd powers of b will be changed from + to; Hence we have,

(a+b)"a"+na"-b+

n(n − 1) qu~22 + n(n − 1) (n − 2) ar−3b3 + &c.

an

2.3

2

.. by addition, (a+b)" + (a−b)" = 2a"+n(n− 1)a”—1 b3 +&c.

n(n−1)(n-2)

by subtraction, (a+b)" — (a—b)" = 2na"-'b+

-

a2-3b3 + &c.

3

n(n-1)(n-2)

or 1⁄2 (a+b)" — — (a−b)"=na"¬1b+

a-3b3+ &c.

2.3

151. Let a=1, b=1, then (a+b)" = (1 + 1)" =2"; and since the several powers of a and b are, in this case, each of

n(n-1)+ n(n-1)(n-2

2

2.3

+

them equal to 1, we have 1+n+
&c.=2′′, i. e. the sum of the coefficients of the nth power
of any binomial is equal to that power of 2 whose index
is n. Thus, for the square, 1+2+1=4=2'; for the cube,

1+3+3+1=8=23; for the fourth power, 1+4+6+4+1 =16=2; &c. &c. If a=1, b=1, in the expression (a−b)", then (1—1)"=0, which shews that the sum of the positive coefficients of (a-b)" is equal to the sum of the negative ones.

XLIII.

On the Expansion of Series.

152. It has already been observed (Art. 149) that when n is a negative number or a fraction, then the series expressing the value of (a+b)" does not terminate. Let n=

m

r

and substitute

m

m m

a+b=a+-a' a+b] =

グ

2

(7)

+ &c. (a)

which is a general expression for finding the approximate value

(*) This series is derived from the preceding one, by resolving the

of any binomial surd quantity, being either positive or ne

m

T

gative, and m and r any whole numbers whatever. (a)

33;

1(1 -3) =

m (m − r ) (m — 2 r ) ( 17 ) = 1 ( 1 −3) (1-6) (20)

2.3

a

&c. &c.

2.3.33

Hence +2=c(1+2+36-&c.)

2.32 C

(a) The proof of the Binomial Theorem, as given in Sect. XLII. goes upon the supposition that ʼn is some positive whole number. To prove the truth of this Theorem when n is negative or fractional, requires a species of analytical investigation not well adapted to an Elementary Treatise. For a general demonstration of it, the Reader is therefore referred to BONNYCASTLE'S Algebra, Vol. II. page 169.