Question

Define Progression.

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Hint: The sequence of the variable and numbers is called series. Series are made in symmetry. A progression is formed by series.

Complete step by step solution: A progression is a series that advances in a logical and predictable pattern. The progression principle states that there is a perfect level of overload in-between a too slow increase and a too rapid increase.
For example:
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.

Note: Students must have a clear concept of progression, so that they can be able to easily differentiate between different types of progressions while solving problems having a particular progression.