Sequences, series, and progressions

Sequences, series and progressions are three mathematical words that we generally consider to imply the same thing: a set of numbers arranged in some definite order. However, there are differences between them which are important from the mathematical point of view.

Sequences

Sequences are a set of numbers, which are arranged according to any specific rule. There is no exception for any type of numbers, any type of rules according to which they are arranged. The set of numbers should have a definite, logical rule according to which they are arranged. It need not be a mathematical formula, but it should be logical. Such a set of numbers are called a sequence of numbers.

For example, the following is a sequence of numbers, because they are arranged according to a definite rule:

{2, 4, 6, 8, 10, 12} Rule: nth term = 2n

The following is a sequence of odd numbers:

{3, 5, 7, 9, 11, 13} Rule: nth term = 2n + 1

The following is also a sequence of numbers, as they too have a logical rule:

{2, 3, 5, 7, 11, 13, 17} Rule: Prime numbers

Series

A series, on the other hand, is a sequence of numbers that is added by + signs. The term 'series' is closely related to the total sum of a sequence of numbers. However, the word 'series' is said to represent the sum of the numbers, and not the sum itself. There is only one difference directly visible between a series and a sequence: The numbers in a series are separated by plus (+) signs, whereas the numbers in a sequence are separated by commas (,).

For example, the following is a series of numbers, because they are separated by + signs, and they are arranged according to a definite rule:

{2+4+6+8+10+12}

The following is a series associated with the sequence of odd numbers:

{3+5+7+9+11+13}

Another important characteristic of a series is that it is always based on a sequence. A series of numbers is always associated with a sequence of numbers.

Progressions

Progressions are yet another type of number sets which are arranged according to some definite rule. The difference between a progression and a sequence is that a progression has a specific formula to calculate its nth term, whereas a sequence can be based on a logical rule like 'a group of prime numbers', which does not have a formula associated with it.

Now you may be wondering that a set of prime numbers should a progression because we can predict its nth term, but a progression needs a specifically stated formula, and, it is to be noted that prime numbers cannot be predicted with the help of any formula; Till date, the formula for the nth prime number has not be found. This means that we can only calculate the nth prime number with the method of selecting each successive number and checking whether it is prime or not.

Therefore {2, 4, 6, 8, 10, 12} represents a progression where the nth term is given by '2n', and, on the other hand, {2, 3, 5, 7, 11, 13, 17} represents a sequence that is not a progression, because although it is based on a definite logical rule (prime numbers), but there is no formula to calculate its nth term.

You may also be wondering why there is no formula for prime numbers till today. The answer is that no one has been able to find the formula for prime numbers. Maybe you can? :)

Coming back to our topic, we can conclude that a sequence is a set of numbers arranged according to some rule, a series is a sequence separated by + signs, and a progression is a sequence that can be stated by a specific mathematical formula like nth term = (2 + n).

We also observe that all series are based on specific sequences, and that although all progressions are sequences, all sequences are not progressions (for example, prime numbers).