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There are many types of progression

1. Arithmetic progression

Arithmetic Progression (AP) is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value.

\[{{\text{N}}^{{\text{th}}}}\]Term of an A.P

The formula for finding the \[{n^{th}}\]term of an AP is:

\[{a_n} = a + \left( {n + 1} \right)\,\, \times \,\,d\]

Where

\[a = First{\text{ }}term\]

\[d{\text{ }} = {\text{ }}Common{\text{ }}difference\]

\[n = number{\text{ }}of{\text{ }}terms\]

\[\;{a_n} = {n^{th}}{\text{ }}term\]

Sum of\[{{\text{n}}^{{\text{th}}}}\]term is given by: \[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right) \times d} \right]\]

Proof: Consider an AP consisting terms having the sequence \[a,{\text{ }}a{\text{ }} + {\text{ }}d,{\text{ }}a{\text{ }} + {\text{ }}2d,{\text{ }} \ldots \ldots .,{\text{ }}a + {\text{ }}\left( {n{\text{ }}-{\text{ }}1} \right){\text{ }} \times {\text{ }}d\]

Sum of first n terms \[ = {\text{ }}a{\text{ }} + {\text{ }}\left( {a{\text{ }} + {\text{ }}d} \right){\text{ }} + {\text{ }}\left( {a{\text{ }} + {\text{ }}2d} \right){\text{ }} + {\text{ }} \ldots \ldots \ldots ..{\text{ }} + {\text{ }}\left[ {a{\text{ }} + {\text{ }}\left( {n{\text{ }}-{\text{ }}1} \right){\text{ }} \times {\text{ }}d} \right]\] …….(i)

Writing the terms in reverse order, we have

\[S{\text{ }} = {\text{ }}\left[ {a{\text{ }} + {\text{ }}\left( {n{\text{ }}-{\text{ }}1} \right){\text{ }} \times {\text{ }}d} \right]{\text{ }} + {\text{ }}\left[ {a{\text{ }} + {\text{ }}\left( {n{\text{ }}-{\text{ }}2} \right){\text{ }} \times {\text{ }}d} \right]{\text{ }} + {\text{ }}\left[ {a{\text{ }} + {\text{ }}\left( {n{\text{ }}-{\text{ }}3} \right){\text{ }} \times {\text{ }}d} \right]{\text{ }} + {\text{ }} \ldots \ldots {\text{ }}\left( a \right)\] …….(ii)

Adding both the equations (i) and (ii)term wise we have:

\[2S{\text{ }} = {\text{ }}\left[ {2a{\text{ }} + {\text{ }}\left( {n{\text{ }}-{\text{ }}1} \right){\text{ }} \times {\text{ }}d} \right]{\text{ }} + {\text{ }}\left[ {2a{\text{ }} + {\text{ }}\left( {n{\text{ }}-{\text{ }}1} \right){\text{ }} \times {\text{ }}d} \right]{\text{ }} + {\text{ }}\left[ {2a{\text{ }} + {\text{ }}\left( {n{\text{ }}-{\text{ }}1} \right){\text{ }} \times {\text{ }}d} \right]{\text{ }} + {\text{ }} \ldots \ldots \ldots .{\text{ }} + {\text{ }}\left[ {2a{\text{ }} + {\text{ }}\left( {n{\text{ }}-{\text{ }}1} \right){\text{ }} \times {\text{ }}d} \right]{\text{ }}\left( {n - terms} \right)\]

\[2S{\text{ }} = {\text{ }}n{\text{ }} \times {\text{ }}\left[ {2a{\text{ }} + {\text{ }}\left( {n{\text{ }}-{\text{ }}1} \right){\text{ }} \times {\text{ }}d} \right]\]

\[S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right) \times d} \right]\]

2. Geometric progression

A geometric sequence or geometric progression (G.P.) is of the form

\[a,\,\,\,ar,\,\,\,a{r^2},\,\,\,a{r^3},\,\,\,\,.......\]

The\[{{\text{n}}^{{\text{th}}}}\]term of a G.P. is

\[{u_n} = a{r^{n - 1}}\]

The sum of\[{\text{n}}\]terms is

\[{S_n} = \dfrac{{a\left( {1 - {r^n}} \right)}}{{1 - r}}\] or \[{S_n} = \dfrac{{a\left( {{r^n} - 1} \right)}}{{r - 1}}\]

3. Harmonic progression

A Harmonic Progression (HP) is defined as a sequence of real numbers which is determined by taking the reciprocals of the arithmetic progression that does not contain\[0\].

The nth term of the Harmonic Progression \[\left( {H.P} \right) = \dfrac{1}{{\left[ {a + \left( {n - 1} \right)d} \right]}}\]

Where

“\[{\text{a}}\]” is the first term of A.P

“\[{\text{d}}\]” is the common difference

“\[{\text{n}}\]” is the number of terms in A.P

The above formula can also be written as:

The\[{{\text{n}}^{{\text{th}}}}\]term of H.P \[ = \dfrac{1}{{\left( {{n^{th}}\,\,term\,\,of\,\,the\,\,corresponding\,\,A.P} \right)}}\]

4. Fibonacci Numbers

The series\[\;2,{\text{ }}4,{\text{ }}6,{\text{ }}8\]is an arithmetic progress. I asked to give the next number, most people would reply$10$. A movement forward, especially one that advances toward some achievement, is called progression.

Geometric progression, Arithmetic, Progression are the type of progression.