We already know that mathematical sequences are sets of numbers arranged in an orderly way.

But what is number series? Are series and mathematical sequences the same? Not necessarily.

Both terms complement each other and are essential for mathematical calculation.

There are certain differences that you should know. Let us understand here the following:

**Differences and similarities between series and sequences:**

The concepts of series and sequences in algebra are often used interchangeably.

However, there is a difference between the series and mathematical sequences. It is best to define both to see it.

Sequences are a set of ordered numbers or values. The most important thing in this list or group is how they are arranged.

Since it classifies them into different types according to their order or what is the limit of the sequence itself.

A pattern is always followed, be it more or less complicated.

The series is a number sequence. But joined by basic mathematical operations like addition.

An important difference between the series and the numerical sequence is that in the series the order is not so relevant.

It is not always a list that follows a certain pattern.

There are two theoretical keys about series and sequences and their differences. But remember that in many books the two concepts can appear as synonyms:

In the sequence, there is always a pattern, in the series not necessarily.

The series is a sum of numbers or sequence, while the sequence is a list or set of values.

**Difference between sequence and progression:**

Progressions are part of the types of sequences that exist in algebra. And are essential to understand the mathematical theory of sequences in the best possible way.

But is succession the same as progression? Not certainly. There are certain differences between succession and progression that you should know as soon as possible.

Why don’t we discover them?

A sequence is an “ordered set of terms that satisfy a certain law”. That is, it is an ordered list of numbers.

A progression is a “succession of numbers or algebraic terms between which there is a law of constant formation”.

What this definition means is that certain types of sequences are called progressions.

That can be defined by giving a rule that relates each term to the next term.

**Sequences and progressions are not the same: **

They both are not the same, but neither can you make a difference between sequence and arithmetic progression.

Since arithmetic progression is one of the types of sequences that exist in algebra only something special.

Furthermore, there is not just one, but two types of progressions: arithmetic and geometric. Let’s start by defining each one:

In algebra, sequences are a set of numbers ordered by a constant pattern with finite and infinite limits.

The progressions are sequences in which the next term of this one is obtained with an addition or a multiplication.

Arithmetic progressions are those in which the next term in the sequence is obtained by adding a fixed number to the previous one. This fixed amount is called the difference (d).

In practice, the word progression is never used in isolation. But is always accompanied by one of these three adjectives: arithmetic, geometric or harmonic.

That is, for practical purposes, a progression is a sequence that meets one of these three conditions:

The difference between two consecutive terms is constant (arithmetic progression).

The quotient between two consecutive terms is constant (geometric progression).

The sequence formed by the inverses (the inverse of the number a is the number 1 / a) of each of the terms in an arithmetic progression (harmonic progression).

For example, {0, 2, 4, 6, 8, …} (the even numbers) is a sequence and, at the same time, an arithmetic progression because the difference between two consecutive terms is always 2.

For example, {2, 3, 5, 7, …} (the prime numbers) is a sequence. But not a progression because it does not satisfy any of the three conditions mentioned above.

For example, we can invent a sequence the n-th term of which is calculated with the formula n2.

That is, {0, 1, 4, 9, 16, …}: it is not a progression either because it does not meet any of the three conditions mentioned above.

**What is the progression?**

An arithmetic progression is an infinite sequence of numbers in which the ratio is constant throughout the entire sequence and is represented by a line.

In other words, an arithmetic progression is a numerical series.

And, therefore, infinite. In which the variation between any two consecutive numbers will always be the same throughout the entire sequence.

**Example:**

Given an arithmetic progression of the form X1, X2,…, X40:

The subscript of the X indicates the position of the number within the sequence. So, there are 40 elements in this progression.

If we had done the calculations, they would be such as:

X2 – X1 = 4 – 1 = 3 ← ratio

X3 – X2 = 7 – 4 = 3 ← ratio

X4 – X3 = 10 – 7 = 3 ← ratio

X39 – X38 = 115 – 112 = 3 ← ratio

X40 – X39 = 118 – 115 = 3 ← ratio.

**What are the different types of progression?**

There are 3 different types of progressions.

Arithmetic Progression (AP)

Harmonic Progression (HP)

Geometric Progression (GP)

**Definition no. 1:** A mathematical sequence in which the difference between 2 consecutive terms is always a constant and it is abbreviated as AP.

**Definition no. 2:** An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms.

The second number is obtained by adding a fixed number to the first one.

**Definition no. 3:** The fixed number which must be added to any term of an AP to get the next term is called the common difference of the AP.