Arithmetic Progression, Geometric Progression and Harmonic Progression are interrelated concepts and they are also one of the most difficult topics in Quantitative Aptitude section of Common Admission Test, CAT. We will discuss them one by one.**Arithmetic Progression (AP)**

The progression of the form: a, a + d, a + 2d, a + 3d … is known as an AP with *first term* = a,and *common difference* = d.

In an AP a, a + d, a + 2d, a + 3d, …, we have:

(i) nth term, T_{n} = a + (n – 1)d

(ii) Sum to n terms, where l is the last term.

(iii) If a, b, c are in AP, then b is called with arithmetic mean (AM) between a andc. In this case, b = (a + c).

(iv) If a, a_{1}, a_{2} … an, b are in AP we say that a_{1}, a_{2} … anare the narithmetic means between a and b.

(v) It is convenient to take:three numbers in AP as (a – d), a, (a + d)

four numbers in AP as (a – 3d), (a – d), (a + d), (a + 3d)**Geometric Progression (GP)**

The progression of the form: a, ar, ar^{2}, ar^{3}, … is known as a GP with first term = a and common ratio = r

(i) nth term, T_{n} = ar^{n– 1}

(ii) Sum to n terms, when r< 1 and when r> 1

(iii) If a, b, c are in GP, then b is the geometric mean (GM) between a andc. In this case, b= √ab .

(iv) If a, a1, a2 … an, b are in GP we say that a1, a2 …an aren geometric means between a and b.

(v) The sum of an infinite GP a, ar, ar^{2}… is .**Harmonic Progression (HP)**

The progression a1, a2, a3… is called an HP if ...is an HP.

If a, b, c are in HP, then b is the harmonic mean between a and c.

In this case, b = **Relationship Between the Means of AP, GP and HP**

If AM, GM and HM be the arithmetic, geometric and harmonic means between a and b, then the following results hold:

Therefore, we can write:

Or GM^{2} = AM x HM........(iv)

Also, we have:

(v)

…which is +ve if a and b are +ve; therefore, the AM of any two +ve quantities is greater than their GM.

Also, from equation (iv) we have, GM^{2} = AM xHM

Clearly then, GM is a value that would fall between AM and HM and from equation (v) it is known that AM > GM, therefore we can conclude that GM > HM.

In words, we can say that the arithmetic, geometric and harmonic means between any two +ve quantities are in descending order of magnitude.

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18 Comment

daman, 1 Year Ago

Plzz explain hp deeply

Ajeet, 1 Year Ago

plz correct the errors in above statements of formula.

Sarthak Sharma, 1 Year Ago

Basic Knowledge of AP GP and HP

kottagaramsravani, 1 Year Ago

please ell me more about HP

Name, 1 Year Ago

Please all discribe tell me about sequence and series

kyisoe, 1 Year Ago

Please tell me more about HP with number

bharath, 1 Year Ago

helpful for me but needed more info about HP

Isha Bhardwaj, 1 Year Ago

cool nfo..huh yeh toh ncert mein bhi de rakha hai..

Shanmukh, 1 Year Ago

THANK YOu for helping

Jaisurya, 1 Year Ago

I was looking for the sum of n terms in a HP. But it is not available here. If the nth term and the sum of n terms are included in this, it will be a use for me

Saurabh Srivastava, 1 Year Ago

This article helps me very much in knowing the relationship between AM,GM,HM, specially. But there is a mistake in GM, i.e., b is GM of a and c if b is equal to root over ac not ab as written in the article

Thiyagarajan, 1 Year Ago

Its useful and make us to know all relation between AP,GP,HP

gousi, 1 Year Ago

hai

arunendra pratap singh, 1 Year Ago

i was looking for HP question but when i turned to this page i came to know about the relatioship between two. its good but very high

smriti, 1 Year Ago

test

Chhanda Halder, 1 Year Ago

What I was really looking for was more information on HP. Unfortunately,the article in this page about HP turned out to be a stub.

anil verma, 1 Year Ago

hi

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Vishnu , 1 Year Ago

Good thanks for the help