Understanding Sequences, Progressions, and Functions in Math

Understanding Sequences, Progressions, and Functions

Sequences

Sequences are unlimited strings of real numbers. Each of the numbers that form a sequence is a term and is designated with a letter and an index that indicates its position in the sequence. The general term is the algebraic expression used to calculate any term, depending on the index.

Recurrent Sequences

Recurrent sequences are those in which terms are defined based on one given earlier, according to a known algebraic expression.

Arithmetic Progressions

A sequence of rational numbers is an arithmetic progression if each term is obtained from the previous one by adding a fixed number, or difference, usually represented by *d*. The general term is: W = A1 + (n-1) * d.

Geometric Progressions

A sequence of rational numbers is a geometric progression if each term is obtained from the previous one by multiplying it by a fixed number, or reason, that is often represented by *r*. The general term is: W = A1 * r^n-1.

Functions

When a relationship of magnitude dependency exists between two states that can be expressed in terms of one magnitude and another, it is called a function. A relationship between two quantities is called a function if the first value carries a single value of the second, which is called the image or transform. The variable that is set beforehand is the independent variable, and the variable inferred from the independent variable is the dependent variable.

Continuous and Discontinuous Functions

A function is continuous if small variations of the independent variable correspond to small variations of the dependent variable; there are no points of discontinuity. A function is discontinuous when jumps occur; the point where the jump occurs is a discontinuity.

Rate of Variation, Increasing, and Decreasing Functions

The rate of variation of a function *f(x)* in an interval [a, b] is the increase or decrease the function experiences when the independent variable changes from value *a* to value *b*. TV[a, b] = f(b) - f(a). A function is increasing in an interval if, for any pair of values in this interval, the rate of variation is positive. A function is decreasing in an interval if, for every pair of values in this interval, the rate of variation is negative.

Maximum and Minimum

A continuous function has a relative maximum at a point if the function is increasing to the left and decreasing to the right. It has a relative minimum if the function is decreasing to the left and increasing to the right.

Linear Functions

Functions of the form y = mx + n are called linear functions; their graph is a straight line.