# §1.2 Elementary Algebra

## §1.2(i) Binomial Coefficients

In (1.2.1) and (1.2.3) and are nonnegative integers and . In (1.2.2), (1.2.4), and (1.2.5) is a positive integer.

1.2.1

### ¶ Binomial Theorem

1.2.2
1.2.3
1.2.4

where is or according as is even or odd.

In (1.2.6)–(1.2.9) and are nonnegative integers and is unrestricted.

1.2.7

## §1.2(ii) Finite Series

### ¶ Arithmetic Progression

1.2.10

where = last term of the series = .

1.2.11.

## §1.2(iii) Partial Fractions

Let be distinct constants, and be a polynomial of degree less than . Then

1.2.12

where

1.2.13

Also,

1.2.14

where

1.2.15

and is the -th derivative of 1.4(iii)).

If are positive integers and , then there exist polynomials , , such that

To find the polynomials , , multiply both sides by the denominator of the left-hand side and equate coefficients. See Chrystal (1959, pp. 151–159).

## §1.2(iv) Means

The arithmetic mean of numbers is

1.2.17

The geometric mean and harmonic mean of positive numbers are given by

1.2.18
1.2.19

If is a nonzero real number, then the weighted mean of nonnegative numbers , and positive numbers with

is defined by

1.2.21

with the exception

1.2.22 and .
1.2.23
1.2.24

For , ,

1.2.25

and

1.2.26

The last two equations require for all .