Question

What is the relation between AP, GP and HP?

Answer 1

Hint: In the given question, we are required to find the relationship between the arithmetic progression, geometric progression and harmonic progression of the same series. So, we will first describe the meaning and representation of all the series and then find the relation of AP, GP and HP of a series.

Complete step-by-step answer:
Arithmetic progressions, geometric progressions and harmonic progressions are the different types of series having their own definitions and properties.
Arithmetic progression or AP is a series of terms or numbers in which the difference between any two consecutive terms is some constant value. So, the series of numbers in arithmetic progression with the first term as a and the common difference between any two consecutive terms as d is given as: $ a,a + d,a + 2d,... $ .
Geometric progression or GP is a sequence or series of numbers in which the ratio between any two consecutive terms is equal. So, the series of numbers in geometric progression with the first term as a and the common ratio as r is given as $ a,ar,a{r^2},.... $ .
Harmonic progression or HP is a sequence or series of numbers in which the difference between the reciprocals of the terms remains constant. In other words, the reciprocal of the terms in arithmetic progression constitutes a harmonic progression. So, the series of numbers in harmonic progression are simply the reciprocals of the terms in arithmetic progression and hence can be represented as: $ \dfrac{1}{a},\dfrac{1}{{a + d}},\dfrac{1}{{a + 2d}},... $ .
Now, we have to find the relation between the arithmetic progression, geometric progression and harmonic progression. If A, G and H are the arithmetic mean, geometric mean and harmonic mean of a series, then we can say that the arithmetic mean is always greater than the geometric mean which in turn, is always greater than the harmonic mean.
So, we have, $ A > G > H $ .
So, the correct answer is “ $ A > G > H $ .”.

Note: We must remember this relationship or inequality between the three types of series as it helps us in solving various different kinds of questions and proving inequalities. We must know the definitions of the types of series before attempting such theoretical questions as it provides the base for such problems.