Arithmetic Progression
Definition:
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the "common difference" and is denoted by $d$.
General Form:
An AP can be represented as: $a,a+d,a+2d,a+3d,\dots $
 $a$ is the first term of the sequence.
 $d$ is the common difference between consecutive terms.
nth Term of an AP:
The nth term of an AP is given by the formula:
${a}_{n}=a+(n1)\times d$
Where:
 ${a}_{n}$ is the nth term of the AP.
 $a$is the first term of the AP.
 $d$is the common difference.
 $n$is the term number.
Sum of First $n$Terms of an AP:
The sum of the first $n$terms of an AP (${S}_{n}$) can be calculated using the formula:
${S}_{n}=\frac{n}{2}\times [2a+(n1)\times d]$
Where:
 ${S}_{n}$ is the sum of the first $n$ terms.
 $n$is the number of terms.
 $a$ is the first term.
 $d$ is the common difference.
Properties of Arithmetic Progression:

Common Difference:
 In an AP, the difference between consecutive terms is constant.

Nth Term:
 The formula to find the nth term of an AP is ${a}_{n}=a+(n1)\times d$.

Sum of an AP:
 The sum of the first $n$ terms (${S}_{n}$) can be calculated using ${S}_{n}=\frac{n}{2}\times [2a+(n1)\times d]$.

Relationship between Terms:
 Any term in an AP can be found by adding the common difference to the previous term.

Finding Number of Terms:
 $n=\frac{{a}_{n}a}{d}+1$can be used to find the number of terms in the AP.
Applications of Arithmetic Progression:
 Mathematics: Formulas for arithmetic means, linear interpolation, etc.
 Finance: Calculating depreciation, amortization, etc.
 Physics: Modelling uniformly changing quantities like distance, velocity, etc.
 Computer Science: Algorithms, series in data structures, etc.