Harmonic progression

Harmonic progression

Let's observe the following two sequences of numbers  {1,3,5,7,...} and {1,1/3,1/5,1/7,...}.

   It is very clear that the first one is an Arithmetic progression (A.P.), and the second one is a sequence of reciprocals of the terms of the first one.

   The second sequence of numbers is known as Harmonic progression (H.P.).

Definition:

  A sequence of numbers {a,b,c,...} are in H.P if and only if the sequence of numbers {1/a, 1/b, 1/c,...} are in A.P.

 Thus, a sequence of numbers forms an H.P. if and only if the sequence of  their  reciprocals are in A.P. 

Therefore, the a sequence of numbers {a,b,c,...} forms a H.P. if the following condition is satisfied.

  Let, {a,b,c,...} are in H.P. Then, {1/a,1/b,1/c,...} are in A.P.

  So, 1/b - 1/a = 1/c - 1/b .

Which is the required condition that  a sequence of number {a,b,c,...} will form a H.P.

In particular, three numbers will form a H.P. , if the ratio of first and third number, is equals to the ratio of the differences between first , second and second, third respectively.

So, {p,q,r} will form a H.P. if , p/r = (p-q)/(q-r).

     But, we should remember that, there is no specific formulas to find the sum of a H.P.

So, to find the sum of a finite number of terms of a H.P. ; we first find the  corresponding A.P. and then using the sum formulas for an A.P. , the required sum is  obtained.

So, the formulas of the A.P.  are also useful for  a H.P.

The n-th term of an A.P.

 If "a" be the first term and "d" be the common difference of an A.P. having the n-th term t(n) , then , 

   t(n)=a+(n-1)d.

The sum of first n terms of an A.P.

   If  "a" , "d", t(n) are the first term, common difference and n-th term of an A.P. respectively, then , the sum of first n terms denoted by s(n) is given by:

  s(n)= (n/2) ×{a+t(n)}

 or, (n/2)×{2a+(n-1)d}, where , we use, t(n)= a+(n-1)d.

Harmonic Mean:

If three numbers are in H.P. then the middle number is called the Harmonic mean.

So, if {x,y,z} are in H.P., then y is called the Harmonic mean of x and z.

 Let, a and b are two numbers and H be their Harmonic mean. 

So, 1/H -  1/a = 1/b - 1/H 

or, 2/H = 1/a + 1/b

or, H= 2ab/(a+b).

Some intersting facts:

(1) If a, b, c are in H.P. then,  1/(bc) , 1/(ac), 1/(ab) are also in H.P.

(2)If a, b, c are in H.P. then, a/(b+c-a) , b/(c+a-b) , c/(a+b-c) are also in H.P.

(3) If a, b, c are in H.P. then, a(b+c) , b(c+a), c(a+b) are in A.P.

(4) If a², b², c² are in A.P. then, (b+c) , (c+a) and (a+b) are in H.P.

(5) If a, b, c are in H.P.  then, a/(b+c), b/(c+a) , c/(a+b) are also in H.P. (a+b+c ≠ 0).

(6) If a, b, c,d are in A.P. then, abc, bcd, abd, acd  are in H.P.

 Note:

Arithmetic- Geometric series:

 Each term of these type of series are expressed as the product of two terms ; one from A.P. and other from G.P.

  The general form of an Arithmetic- Geometric progression is, 

 a×1 + (a+d)×r + (a+2d)×r²+...

To find the sum of an Arithmetic - Geometric progression  a special method is used.

An example of an Arithmetic- Geometric progression is given by:

 1+2a+ 3a² + 4a³+..., a not equals to 1.

  If you find out any incorrect information or know anything more about this , please write it in the comment section!