The value of a circulating decimal is a fraction whose numerator is the period, and denominator a number consisting of as many nines as there are digits in the period.

(W. 225; F. 50_2.)

(23.) Any three of the five quantities az, ano ?, n, s, being given, the other two may be found from the following formulæ.

- ai s-an

(a, a, 8);

log an - log a,

+1. log (8-a,) - log (8-an)

a =ran- (r – 1)s. (am r, 8); log a, - log (ra, -1-1.s)

logr

(G. p. 281.) PERMUTATIONS AND COMBINATIONS. (24.) The number of permutations of n things taken r at

n =

+1.

a time

= n(n-1)(n-2)......(n-p+1). B.

n(n-1)(n-2)...(n-1+1) The number of combinations

1. 2 3 (W. 226_31; E. 352—60; Bour. 146—50.)

......

a The equations containing logarithms, are placed here on account of their connexion with the rest, although the properties of logarithms have not yet been stated.

® The number of permutations = (n.

7

n

m

BINOMIAL THEOREM.

n(n-1) (25.) (a + b)" =a" + an - 16+ a" -%% + &c.

1

1 2 the m + 1 th term being (m - 1) (x – 2)....(n – m +1)

an-mbm. 1 . 2 . 3 If n is a positive integer, then

the number of terms n +1;

the coefficients of terms equally distant from both extremities of the series are equal ;

the coefficient of every term is an integer ;

also the theorem may be put under the following forms:
(a + b)"
a”

an-16
1.2.3...n 1.2.3...n 1.2.3...(n-1).1

+

The product of n simple factors, on which the binomial theorem depends, may be thus expressed :

P. (*+a)=8--+"C.C.).

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

an – 16" - (a? + b) 1 2

n

2n-1 (a + b)2n-1=(aan – ? + b2n – 1) +

ab(qon-s+b?n5) 1

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

a’b?(q?n - 5

+62n–5) + &c. 1 . 2

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

Fan-181 - (a + b). • 1

2 ...(n-1)

2n

? (a + b)2n = Sim

m-1

(ab)-1. (@2n – 2m +2

+ hen - 2m+2) +

+2)+(ab)".

m

-1

n

3

a

+ &c. 8

1(1+n)(1 + 2n) -- 3 6
1.2 3

(W. 232–8; G. Chap. xii; Bour. 182—90.)

n

.