Question

Define Progression.

Answer 1

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
$$a=Firstterm$$
$$d=Commondifference$$
$$n=numberofterms$$
$$a_n=n^{th}term$$
Sum of$${\text{n}}^{\text{th}}$$term is given by: $$S_n={\displaystyle \frac{n}{2}}\left[2a+\left(n-1\right)\times d\right]$$
Proof: Consider an AP consisting terms having the sequence $$a,a+d,a+2d,\dots \dots .,a+\left(n-1\right)\times d$$
Sum of first n terms $$=a+\left(a+d\right)+\left(a+2d\right)+\dots \dots \dots ..+\left[a+\left(n-1\right)\times d\right]$$ …….(i)
Writing the terms in reverse order, we have
$$S=\left[a+\left(n-1\right)\times d\right]+\left[a+\left(n-2\right)\times d\right]+\left[a+\left(n-3\right)\times d\right]+\dots \dots \left(a\right)$$ …….(ii)
Adding both the equations (i) and (ii)term wise we have:
$$2S=\left[2a+\left(n-1\right)\times d\right]+\left[2a+\left(n-1\right)\times d\right]+\left[2a+\left(n-1\right)\times d\right]+\dots \dots \dots .+\left[2a+\left(n-1\right)\times d\right]\left(n-terms\right)$$
$$2S=n\times \left[2a+\left(n-1\right)\times d\right]$$
$$S={\displaystyle \frac{n}{2}}\left[2a+\left(n-1\right)\times d\right]$$

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
$$S_n={\displaystyle \frac{a\left(1-r^n\right)}{1-r}}$$ or $$S_n={\displaystyle \frac{a\left(r^n-1\right)}{r-1}}$$

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 $$\left(H.P\right)={\displaystyle \frac{1}{\left[a+\left(n-1\right)d\right]}}$$
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 $$={\displaystyle \frac{1}{\left(n^{th}termofthecorrespondingA.P\right)}}$$

4. Fibonacci Numbers
The series$$2,4,6,8$$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.