# Mathematical Progressions

A mathematical progression is a sequence of numbers such that all the members of the sequence are governed by certain rule, which defines the relation between consecutive terms. There are three types of mathematical progressions:

#### 1. Arithmetic Progression

An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the **difference** between the consecutive terms is constant.

For an AP with common difference as **d** and first term as $a_1$, the terms of the progression can be represented as: $a_1, a_1+d, a_1+2d, a_1+3d … a_1+(n-1)d$

Consider the following sequence for example:

**{1,2,3,4,5}**: AP with common difference as**1****{2,4,6,8,10}**: AP with common difference as**2****{5,10,15,20,25}**: AP with common difference as**5**

For a sequence of terms ${a_1, a_2, … . a_n}$ with common difference as **d**

- The $n^{th}$ term is given by : $a_n = a_1 + (n-1)d$
- Sum of first n terms of the sequence $S_n = \frac{n}{2}(a_1+a_n)$
- The sum of first
**n**terms can also be represented as $S_n = \frac{n}{2}\bigl(2a_1 + (n-1)\bigr)d$ - $N^{th}$ term can be represented as $Sum(n) - Sum(n-1)$
- The middle term of three consecutive terms of an AP is the mean of the first and last. $a_2 = \frac{a_1 + a_3}{2}$

#### 2. Geometric Progression

A geometric progression (GP) or geometric sequence is a sequence of numbers such that the **ratio** between the consecutive terms is constant.

On similar lines as mentioned above for AP, consider the first term to be **a** and common ratio to be **r**, the terms of GP can be represented as: $a, ar, ar^2, ar^3 … ar^{n-1}$

Consider the following sequence for example:

**{2,4,8,16,32}**: GP with first term as**2**and common ratio as**2****{5,10,20,40,80}**: GP with first term as**5**and common ratio as**2**

Some useful tips on GP:

- The $n^{th}$ term is given by : $a_n = ar^{n-1}$
- Sum of first
**n**terms $S_n = \frac{ar^n - 1}{r-1}; r \ne 1$

#### 3. Harmonics Progression

Harmonic Progression or harmonic series is a sequence of numbers that originally form an arithmetic progression. Consider the terms $a, a+d, a+2d … a+(n-1)d$ of an arithmetic progression with first term as **a** and common difference as **d**, the terms of the HP will be given by:
$$\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}… \frac{1}{a+(n-1)d}$$

Consider the following examples:

- $1,\frac{1}{2}, \frac{1}{3},\frac{1}{4},\frac{1}{5}$;
**{1,2,3,4,5}**forms an AP with`a=1`

and`d=1`

- $\frac{1}{2},\frac{1}{4},\frac{1}{6},\frac{1}{8},\frac{1}{10}$;
**{2,4,6,8,10}**forms an AP with`a=2`

and`d=2`

- $\frac{1}{5},\frac{1}{10},\frac{1}{15}, \frac{1}{20},\frac{1}{25}$;
**{5,10,15,20,25}**forms an AP with`a=5`

and`d=10`