Harmonic Progression: Overview, Questions, Preparation

When you arrange a series of numbers into a predictable pattern, it is termed progression. Progression is a sequence of numbers that follow particular rules. Progression is can be of three types, i.e. Arithmetic Progression, Geometric Progression & Harmonic Progression. 

For solving the problems related to harmonic progression, firstly perceive the corresponding sum of an arithmetic progression. In simple terms, the nth term of the progression must be equal to the nth terms reciprocal of the related Arithmetic Progression. 

Harmonic Progression Formula

Harmonic Progression’s n^th term= 1/ [a+(n-1)d]

Where:

“a” depicts the first-term of the Arithmetic Progression.

“d” represents the common difference.

“n” depicts overall no. of terms in Arithmetic Progression.

If there is a harmonic expression, i.e. 1/a, 1/a+d, 1/a+2d, …., 1/a+(n-1), the actual formula for finding the sum of n terms is:

Sn=1/d(ln){2a+(2n−1)d/2a−d}

In which,

“a” is first term

“d” is difference

“ln”: natural logarithm

To understand the working of harmonic progression, let’s go through some solved examples: 

1. Determine the 4th and 8th term of the harmonic progression 6, 4, 3,…

Solution.

Harmonic progression = 6, 4, 3

Find the corresponding arithmetic progression for the H.P.

Arithmetic Progression = ⅙, ¼, ⅓, ….

By A.P. T2 -T1 is equal to T3 -T2 
i.e. 1/12 = d

For finding the 4^th term of the given A. P, the formula is: 

n^th term = a+(n-1)d

Where, a = ⅙ and d= 1/12

For 4^th term n=4

Putting these values in the formula:

4^th term = (⅙) +(4-1)(1/12)

=> (⅙)+(3/12)

=> 5/12

Repeat the same steps for 8^th term

8^th = (⅙) +(8-1)(1/12)

= (⅙)+(7/12)

= 9/12

As we know, harmonic progression is reciprocal, I.e. reverse of an Arithmetic Progression:

Thus, the 4^th term becomes 1/4^th term = 12/5 and

8^th term becomes 1/8^th term = 12/9 or 4/3

2. If the first two terms of a harmonic progression are 1/16 and 1/13, find the maximum partial sum?

Solution.

Harmonic series will be: 1/16, 1/13, 1/10,1/7, 1/4, 1/1, 1/-2, 1/-5

Maximum partial sum for this series will be 1/16 + 1/13 + 1/10 +1/7 + 1/4 + 1/1 = 1.63

3. Compute the 16^th term of H.P. if the 6th and 11^th term of H.P. is 10 and 18.

Solution.

Write H.P. in terms of A.P.:

6^th term of Arithmetic Progression= a+5d = 1/10 [Equation 1]

11^th term of Arithmetic Progression= a+10d = 1/18 [Equation 2]

Solve equation 1 and 2 for getting the values of a and d:

a =13/90

d = -2/ 225

For finding 16^th term, write expression as:

a+15d 

= (13/90) – (2/15) 

= 1/90

The 16^th term of the Harmonic Progression is equal to 1/16th term of an Arithmetic Progression that is 90

Thus, the 16^th term is 90.

Q: What is the formula of Harmonic Mean?

Q: What are the four types of sequences?

Q: Is harmonic progression in JEE mains?

Q; Why do we calculate the harmonic mean?

Q; How do you find the common difference in harmonic progression?