Question

What are the types of progression?

Answer 1

Hint: To answer the type of progression, we should first know what progression and series. The sequence of the variable and numbers is called a series. We know that series are made in symmetry and a progression is formed by series. We can say that a progression is a series that advances in a logical and predictable pattern.

Complete step by step answer:
We know that the progression principle states that there is a perfect level of overload in between a too slow increase or a too rapid increase. There are mainly $3$ types of progression:
Arithmetic progression: Arithmetic progression or AP is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value.
The formula for finding the $n^{th}$ term of an AP is:
$a_n=a+(n-1)d$,
Where we know that $a$ is the first term, $d$ is the common difference and $n=$number of terms.
$a_n=n^{th}term$
We can find the sum of the $n^{th}$ term of AP by the formula:
$S_n={\displaystyle \frac{n}{2}}\left[2a+\left(n-1\right)d\right]$

Geometric progression: We know that geometric progression or GP is progression of numbers with a constant ratio between each number and the one before. The general form of a geometric progression is of the form;
$a,ar,a{r}^2,a{r}^3...$
We can calculate the $n^{th}$ term of G.P with the formula:
$a_n=a{r}^{n-1}$
where $a$ is the first term and $r$ is the ratio between them.
We can calculate the sum of $n$ terms with the formula:
$S_n={\displaystyle \frac{a(1-r^n)}{1-r}}$
$\Rightarrow S_n={\displaystyle \frac{a(r^n-1)}{r-1}}$

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 zero. The formula of the $n^{th}$ term of the Harmonic progression is:
${\displaystyle \frac{1}{a+(n-1)d}}$,
where $a$ is the first term, $d$ is the common difference and $n$ is the number of terms in AP.
We can also write the above formula as:
${\displaystyle \frac{1}{n^{th}termofthecorrespondingA.P}}$ .
Hence these are the different types of A.P.

Note: We should note that the numbers $2,4,6,8$ is an arithmetic progression.These are called the Fibonacci numbers. The Fibonacci sequence is a series of numbers where a number is an addition of the last two numbers starting with $0$ and $1$ . The Fibonacci sequence can be written as:
$0,1,1,2,3,5,8,13,21,34,55...$
We can write the rule of this expression as:
$X_n=X_{n-1}+X_{n-2}$