Harmonic Progression: Examples (with Solutions), Algebra, Quantitative Aptitude | CSAT Preparation - UPSC PDF Download

Harmonic Progression (H.P.)

1. A series of quantities is said to be in a harmonic progression when their reciprocals are in arithmetic progression.
    
2. e.g.    are in HP as their reciprocals
    
3. 3, 5, 7, …, and a, a + d, a + 2d….…. are in AP.

nth term of HP

1. Find the nth term of the corresponding AP and then take its reciprocal.
    
2. If the HP be 
    
3. Then the corresponding AP is a, a + d, a + 2d, ……
    
4. T_n of the AP is a + (n - 1) d
5. T_nth of the HP is …… 
    
6. In order to solve a question on HP, one should form the corresponding AP.

A comparison between AP and GP

Arithmetic - Geometric progression
a + (a + d)r + (a + 2d)r² + (a + 3d)r³ + ……………. Is the form of Arithmetic geometric progression (A.G.P).
One part of the series is in Arithmetic progression and other part is a Geometric progression.

Arithmetic geometric series can be solved as explained in the example below:

Relation between AM, GM and HM:

For two positive numbers a and b

This mean A, G, H are in G.P.
Verifying for numbers 1, 2

Hence  

Toolkit

Ex.1 Find the sum of 1 + 2x + 3x² + 4x³ + … ∞

Sol. The given series in an arithmetic-geometric series whose corresponding A.P. and G.P. are 1, 2, 3, 4,…
and 1, x, x², x³, … respectively. The common ratio of the G.P. is x. Let S_∞ denote the sum of the given
series.

Ex.2 If the first item of an A.P is 12, and 6th term is 27. What is the sum of first 10 terms?

Sol.

Ex.3 If the fourth & sixth terms of an A.P are 6.5 and 9.5. What is the 9th term of that A.P?
 

Sol.

Ex.4 What is the arithmetic mean of first 20 terms of an A.P. whose first term is 5 and 4th term is 20?

Sol.

Ex.5 The first term of a G.P is half of its fourth term. What is the 12th term of that G.P, if its sixth term
 is 6

Sol.

Ex.6 If the first and fifth terms of a G.P are 2 and 162. What is the sum of these five terms?

Sol.

Ex.7 What is the value of r + 3r² + 5r³ + - - - - -

Sol. 

Ex.8 The first term of a G.P. 2 and common ratio is 3. If the sum of first n terms of this G.P is greater
 than 243 then the minimum value of ‘n’ is

Sol.

Ex.9 

Sol.

Ex.10  

Sol.