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:
${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$

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:
${S_n} = \dfrac{{a(1 - {r^n})}}{{1 - r}}$
$\Rightarrow {S_n} = \dfrac{{a({r^n} - 1)}}{{r - 1}}$

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:
$\dfrac{1}{{a + (n - 1)d}}$,
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:
$\dfrac{1}{{{n^{th}}\,term\,of\,the\,corresponding\,A.P}}$ .
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:
${X_n} = {X_{n - 1}} + {X_{n - 2}}$