Harmonic Progression  

Definition

The Harmonic Progression (HP) is defined as a sequence of real numbers which is determined by taking the reciprocals of the arithmetic progression that does not contain 0. 

For example

Let a ,a+ d , a+ 2d , …….be an arithmetic progression , then the harmonic progression can be written as  

                1/a , 1/(a +d) , 1/(a +2d) ,……          where a is the first term and d is the common difference of the corresponding AP. 

Important Note  

If a , b ,c is in HP , then  

                                         a/c = (a -b)/(b -c) 

n^th Term of the HP

The n^th term of the HP is equal to the reciprocal of the n^th term of the AP. 

Therefore , n^th term of the HP = 1/n^th term of the corresponding AP 

                                                       = 1/[ a+(n-1)d] 

Where a is the first term and d is the common difference of the AP . 

Harmonic Mean (HM)

The Harmonic Mean between two numbers a and b is given by  

         H = 2ab / (a+ b) 

Let a , H₁ ,H₂ ,…….,H_n , b be a harmonic progression, then H₁ ,H₂ ,…..,H_n are called n harmonic means between a and b . 

Therefore ,     H_n = ab(n +1) /( na +b) 

Relationship between AM , GM and HM  between two numbers

Let A , G and H be the AM ,GM and HM between two numbers a and b , then  

         AH = G² 

                 Also , A > G > H 

Sum of n terms of the HP

Let 1/a , 1/a +d , 1/a +2d , …., 1/a + (n-1)d be in HP , then  

Where a and d are the first term and common difference in the corresponding AP. 

Solved Examples

(1)Which term of the given HP 1/10,1/8,1/6,1/4,…..is -1/26 ? 

Sol-

Let -1/26 be the n^th term of the HP , then n^th term of the AP  

      10,8,6,4,….is -26 

                               with a = 10 and d = -2 

Then , -26 = 10 + (n-1)×(-2) 

         Or    n = 19 

Therefore, -1/26 is the 19^th term of the given HP . 

(2)If , 1/12 and 1/42 are the 4^th and 13^th term of any HP , then find n^th term of the given HP . 

Sol.     

Let a and d be the first term and common difference of the corresponding AP . 

Then , a +3d =12 and a +12d = 42 

Solving these two , we get , a = 2 and d =10/3 

Thus , n^th term of the AP = a +(n-1)d 

                                              = 2 + (n-1)×10/3 

                                              = (10n -4)/3 

Therefore , n^th term of the HP = 3 /(10n -1) . 

(3)Find the value of HM of 1 and 9?

Sol. 

Let H be the HM of 1 and 9. 

Then , H = 2ab/(a +b) 

                = 2×1×9/(1+9) 

                  = 18/10 =9/5 

(4)If in any HP , n^th term is m and m^th term is n , then find the value of (m+ n)^th term of the HP? 

Sol. 

 Let a and d be the first term and common difference of the corresponding AP . 

Then , 1/n = a +(m-1)d     and 1/m = a + (n-1)d 

Solving , these two we get , a = 1/mn , d = 1/mn 

Thus , (n+ m)^th term of the AP = a +(m +n-1)d = (m +n)/mn 

Therefore, (m +n)^th term of the HP = mn/(m +n)