# 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:

${a}_{n}=a+\left(n-1\right)×d$

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:

${S}_{n}=\frac{n}{2}×\left[2a+\left(n-1\right)×d\right]$

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:

1. Common Difference:

• In an AP, the difference between consecutive terms is constant.
2. Nth Term:

• The formula to find the nth term of an AP is ${a}_{n}=a+\left(n-1\right)×d$.
3. Sum of an AP:

• The sum of the first $n$ terms (${S}_{n}$) can be calculated using ${S}_{n}=\frac{n}{2}×\left[2a+\left(n-1\right)×d\right]$.
4. Relationship between Terms:

• Any term in an AP can be found by adding the common difference to the previous term.
5. 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.