# Algebra

## Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers

For two numbers x and y, let x, a, y be a sequence of three numbers. If x, a, y is an arithmetic progression then 'a' is called *arithmetic mean*. If x, a, y is a geometric progression then 'a' is called *geometric mean*. If x, a, y form a harmonic progression then 'a' is called *harmonic mean*.

Let AM = arithmetic mean, GM = geometric mean, and HM = harmonic mean. The relationship between the three is given by the formula

Below is the derivation of this relationship.

## Derivation of Sum of Finite and Infinite Geometric Progression

**Geometric Progression, GP**

Geometric progression (also known as geometric sequence) is a sequence of numbers where the ratio of any two adjacent terms is constant. The constant ratio is called the common ratio, r of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by r.

**Eaxamples of GP:**

- 3, 6, 12, 24, … is a geometric progression with r = 2
- 10, -5, 2.5, -1.25, … is a geometric progression with r = -1/2

## Derivation of Sum of Arithmetic Progression

**Arithmetic Progression, AP**

Definition

*d*.

Examples of arithmetic progression are:

- 2, 5, 8, 11,... common difference = 3
- 23, 19, 15, 11,... common difference = -4

**Derivation of Formulas**

Let

## Sum and Product of Roots

The quadratic formula

give the roots of a quadratic equation which may be real or imaginary. The ± sign in the radical indicates that

where x_{1} and x_{2} are the roots of the quadratic equation ax^{2} + bx + c = 0. The sum of roots x_{1} + x_{2} and the product of roots x_{1}·x_{2} are common to problems involving quadratic equation.

## Derivation of Quadratic Formula

The roots of a quadratic equation *ax*^{2} + *bx* + *c* = 0 is given by the quadratic formula

The derivation of this formula can be outlined as follows:

- Divide both sides of the equation
*ax*^{2}+*bx*+*c*= 0 by*a*. - Transpose the quantity
*c*/*a*to the right side of the equation. - Complete the square by adding
*b*^{2}/ 4*a*^{2}to both sides of the equation. - Factor the left side and combine the right side.
- Extract the square-root of both sides of the equation.
- Solve for
*x*by transporting the quantity*b*/ 2*a*to the right side of the equation. - Combine the right side of the equation to get the quadratic formula.

See the derivation below.