Question

What is AP GP and HP?

Answer 1

To get the idea of the AP GP and HP observe the pattern the series is following. Check the pattern in the starting four to five terms after observing them you come to know what pattern series is following and then check the condition of all the series AP GP and HP. By doing this you are able to know which type of series is given.

Complete step by step answer:
All the three terms AP GP and HP are the sequence i.e. a kind of series. Let us discuss all the three terms one by one in brief
Arithmetic Progression (AP): Arithmetic Progression or AP is a sequence or a series of numbers in which the difference between the two consecutive terms is some constant value. For example, consider the series of numbers as ${{a}_{1}},{{a}_{2}},{{a}_{3}},{{a}_{4}},{{a}_{5}},...$ now if we want to check the series is an AP or not then we find the common difference between the consecutive numbers if they are equal then we can say that series is an AP series.
If the series follows ${{a}_{2}}-{{a}_{1}}={{a}_{4}}-{{a}_{3}}$ this means that series is in AP. If we wish to find the $n^{th}$ term of an AP then the formula used is,
${{a}_{n}}={{a}_{1}}+(n-1)d$
Where,
${{a}_{n}}=n^{th}$ term of the AP
${{a}_{1}}=$ First term of the AP
$n=$ Total number of terms
$d=$ Common difference
Geometric Progression (GP): Geometric Progression or GP is a sequence or a series of numbers in which the ratio between the two consecutive terms is the same. For example, consider the series of numbers as ${{a}_{1}},{{a}_{2}},{{a}_{3}},{{a}_{4}},{{a}_{5}},...$ now if we want to check the series is in GP or not then we find the common ratio between the consecutive numbers if they are equal then we can say that series is a GP series.
If the series follows $\dfrac{{{a}_{2}}}{{{a}_{1}}}=\dfrac{{{a}_{4}}}{{{a}_{3}}}$ this means that series is in GP. If we wish to find the $n^{th}$ term of a GP then the formula used is,
${{a}_{n}}={{a}_{1}}{{r}^{n-1}}$
${{a}_{n}}=n^{th}$ term of the GP
${{a}_{1}}=$ First term of the GP
$r=$ Common ratio
$n=$ Number of term
Harmonic Progression (HP): The harmonic progression or HP is a sequence of the numbers formed by taking the reciprocals of the arithmetic progression.
The HP series can be represented as,
$\dfrac{1}{a},\dfrac{1}{a+d},\dfrac{1}{a+2d},\dfrac{1}{a+3d},\dfrac{1}{a+4d},...$

Note: There is a relationship between AP GP and HP. And by the definition we can say that arithmetic progression is always greater than or equal to harmonic progression. If $a,b$ and $c$ are the mean of AP GP and HP respectively then the relationship between them are ${{b}^{2}}=ac$ and one more relation between there mean are $a > b > c$ .