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:
…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.