# Smallest number for given remainders

# Arithmetic, geometric, and harmonic progressions

**Elements**

a_{1} = value of the first term

a_{m} = value of any term after the first term but before the last term

a_{n} = value of the last term

n = total number of terms

m = m^{th} term after the first but before n^{th}

d = common difference of arithmetic progression

r = common ratio of geometric progression

S = sum

## Arithmetic Progression, AP

Arithmetic progression is a sequence of numbers in which the difference of any two adjacent terms is constant. The constant difference is commonly known as common difference and is denoted by d. Examples of arithmetic progression are as follows:

# Example 04 - Simultaneous Non-Linear Equations of Three Unknowns

**Problem**

Solve for x, y, and z from the following system of equations.

$x(y + z) = 12$ → Equation (1)

$y(x + z) = 6$ → Equation (2)

$z(x + y) = 10$ → Equation (3)

# Example 03 - Simultaneous Non-Linear Equations of Three Unknowns

**Problem**

Find the value of x, y, and z from the given system of equations.

$x(x + y + z) = -36$ → Equation (1)

$y(x + y + z) = 27$ → Equation (2)

$z(x + y + z) = 90$ → Equation (3)

# Example 02 - Simultaneous Non-Linear Equations of Three Unknowns

**Problem**

Find the value of x, y, and z from the following equations.

$xy = -3$ → Equation (1)

$yz = 12$ → Equation (2)

$xz = -4$ → Equation (3)

# System of Equations

**System of Linear Equations**

The number of equations should be at least the number of unknowns in order to solve the variables. System of linear equations can be solved by several methods, the most common are the following,

1. Method of substitution

2. Elimination method

3. Cramer's rule

Many of the scientific calculators allowed in board examinations and class room exams are capable of solving system of linear equations of up to three unknowns.

# Example 01 - Simultaneous Non-Linear Equations of Three Unknowns

**Problem**

Solve for x, y, and z from the following simultaneous equations.

$z^x \, z^y = 100\,000$ → equation (1)

$(z^x)^y = 100\,000$ → equation (2)

$\dfrac{z^x}{z^y} = 10$ → equation (3)

# Example 03 - Sum and product of roots of quadratic equation

**Problem**

Find the sum and product of roots of the quadratic equation x^{2} - 2x + 5 = 0.

# Example 02 - Quadratic equation problem

**Problem**

Determine the equation whose roots are the reciprocals of the roots of the equation 3x^{2} - 13x - 10 = 0.