Introduction
Progression is defined as a list of numbers that exhibit a specific pattern. It is also known as a sequence. Every (n+1)^{th} term in a progression is obtained by applying a specific rule on its n^{th} term. For example, 2, 4, 6, 8, ... is a progression because every number is obtained by adding 2 to its previous number. This pattern can vary from progression to progression.
In general, every progression term is defined by the n^{th} term, denoted by a_{n}. For example, the n^{th} term of a progression 2, 4, 6, 8, ... is a_{n} = 2n. Substituting n = 1, 2, 3, ... here, we will get the 1^{st}, 2^{nd}, and 3^{rd}, .... terms.
In mathematics, there are mainly three types of progressions:
 Arithmetic progressions
 Geometric progressions

Harmonic progressions
This blog will discuss one of the types of progression,i.e., Arithmetic Progression.
Arithmetic Progression
An arithmetic progression is a sequence in which the difference between any two consecutive terms is constant. This constant difference in the arithmetic progression is known as the common progression difference and is generally denoted by â€˜dâ€™.
Common Difference
The common difference is the difference between any two consecutive terms that can be easily calculated using the formula d = a_{n+1 } a_{n}. In an A.P., this difference remains constant throughout the entire sequence.
Consider the following examples:

Given a sequence: 1, 3, 7, 11, 15,...
To calculate the common difference of the given sequence we follow the formula d= a_{2 } a_{1}.i.e., 3  (1) = 4. Hence, the common difference in the given sequence is 4. Since this difference is the same among the entire sequence, the given sequence is an A.P. An increasing A.P. is a type of progression where the common difference is positive.

Given a sequence: 25, 20, 15, 10, 5,...
To calculate the common difference of the given sequence, we follow the formula d= a_{2 } a_{1}.i.e., 20  (25) = 5. Hence, the common difference in the given sequence is four. Since this difference is the same among the entire sequence, the given sequence is an A.P. A type of progression where the common difference is a negative number is called decreasing A.P.
General Term
An arithmetic progression is generally represented as a_{1}, a_{2}, a_{3},...., a_{n}. If the first term, generally denoted by â€˜aâ€™, and the common difference â€˜dâ€™ in any given arithmetic sequence is known, we can easily calculate the n^{th} term using the given formula.
a_{n} = a + (n1)*d
Here, a_{n} is known as the general term of the sequence. We can easily find the complete sequence if the general term for a sequence is known.
Derivation of the General Term
We have learned earlier that the common difference of an arithmetic sequence is d= a_{n+1 } a_{n}.
The given expression can also be written as a_{n+1 }= a_{n }+ d. Substitute n=1, 2, 3, â€¦
The first term is represented by a and the common difference by d. The sequence will be a, a+d, a+ 2*d, â€¦.. Following the pattern, the n^{th} term will become a+ (n1)*d.
Examples

Given an A.P. whose fifth term is 34, the ninth term is 50. Find the first term of the sequence, the common difference, and the n^{th} term.
The fifth term of an A.P. is represented by a+ 4d = 34
The ninth term of an A.P. is represented by a+ 8d = 50
Solving the above two equations will give a = 18 and d= 4.
The n^{th} term for the A.P. will be a+ (n1)*d. I.e., 18+ (n1)*4.

Given an A.P. whose first term is 2, the common difference is 2. Find the A.P.
An A.P. is represented as a, a+d, a+2*dâ€¦ Substituting the value of a and d in the above sequence will give us the desired A.P. The A.P. is 2, 0, 2, 4, â€¦