Question

What are the types of progression?

Answer 1

Types of Progression:

Progression is a series of numbers that follow a certain pattern.

• Arithmetic progression:

It is a kind of progression in which the difference between consecutive terms of the series are equal.

If $a_1,a_2,a_3,a_{4,}{a}_5,..........$ are in arithmetic progression then, $a_2-a_1=a_3-a_2=a_4-a_3=.........$

The formula for an arithmetic progression to find $n^{th}$term is $a_n=a+(n-1)d$, where $a$is the first term of the series, $d$is the common difference, $n$is the number of terms, and $a_n$is the $n^{th}$ term.

• Geometric progression:

This is a series of numbers in which the ratio of consecutive elements is equal.

If $a_1,a_2,a_3,a_{4,}{a}_5,..........$ are in geometric progression then, $\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=......$

The series can be rewritten as $a,ar,a{r}^2,a{r}^3,a{r}^4,..........$where $r$is the common ratio.

The formula for a geometric progression to find $n^{th}$term is $a_n=a{r}^{n-1}$, where $a$is the first term of the series, $n$is the number of terms, and $r$is the common ratio.

• Harmonic progression:

This is a series in which the reciprocal of each of the terms in the series is in arithmetic progression.

The general form of a harmonic series is $\frac{1}{a},\frac{1}{a+d},\frac{1}{a+2d},\frac{1}{a+3d}..........$

Here $a,a+d,a+2d,a+3d,\cdots$ are in AP.

The $n^{th}$ term of the AP is $t_n=a+\left(n-1\right)d$.

The formula for the $n^{th}$term of a harmonic progression is $H_n=\frac{1}{a+(n-1)d}$.where $a$is the first term of the AP series, $d$is the common difference of the AP series, $n$is the number of terms, and $H_n$is the $n^{th}$ term.