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Arithmetic Progression (AP) Geometric (GP) and Harmonic Progression (HP): CAT Quantitative Aptitude

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, Tn = 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, a1, a2 … an, b are in AP we say that a1, a2 … 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, ar2, ar3, … is known as a GP with first term = a and common ratio = r
(i) nth term, Tn = arn– 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, ar2… 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 GM2 = 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,  GM2 = 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

Vishnu ,

Good thanks for the help

daman,

Plzz explain hp deeply

Ajeet,

plz correct the errors in above statements of formula.

Sarthak Sharma,

Basic Knowledge of AP GP and HP

kottagaramsravani,

please ell me more about HP

Name,

Please all discribe tell me about sequence and series

kyisoe,

Please tell me more about HP with number

bharath,

helpful for me but needed more info about HP

Isha Bhardwaj,

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

Shanmukh,

THANK YOu for helping

Jaisurya,

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,

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,

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

gousi,

hai

arunendra pratap singh,

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,

test

Chhanda Halder,

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,

hi

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