# Harmonic Progression

A series of non-zero numbers is said to be harmonic progression
(abbreviated H.P.) if the series obtained by taking reciprocals of the
corresponding terms of the given series is an arithmetic progression.

For example, the series 1 +1/4 +1/7 +1/10 +..... is an H.P. since the series
obtained by taking reciprocals of its corresponding terms i.e. 1 +4 +7 +10
+... is an A.P.

A general H.P. is 1/a + 1/(a + d) + 1(a + 2d) + ...

nth term of an H.P. = 1/[a +(n -1)d]

Three numbers a, b, c are in H.P. iff 1/a, 1/b, 1/c are in A.P.

i.e. iff 1/a + 1/c = 2/b

i.e. iff b= 2ac/(a + c)

Thus the H.M. between a and b is H = 2ac/(a + c)

## Illustrative Examples

### Example

The 7th term of an H.P. is 1/10 and 12th term is 1/25 Find the 20th term, and the nth term.

### Solution

Let the H.P. be 1/a + 1/(a + d) + 1(a + 2d) + ...

The 7th term = 1/(a + 6d) = 1/10 => a +6 d = 10

The 12th term = 1/(a + 11d) = 1/25 => a +11 d = 25

Solving these two equations, a = -8, d = 3

Hence 20th term = 1/(a+19d) = 1/[-8 + 9(3)] = 1/49

and nth term = 1/[a +(n -1)d] = 1/[-8 +(n -1) 3] = 1/[3n - 11]

### Example

Prove that three quantities a, b, c are in A.P., G.P., or H.P. iff

(a-b)/(b-c) = a/a, a/b or a/c respectively.

### Solution

a, b, c are in A.P. iff b -a = c -b i.e.

iff (a-b)/(b-c) = 1 = a/a

Also a, b, c are in G.P.

iff b/a = c/b i.e iff 1 - b/a = 1 - c/b

i.e. iff (a-b)/a = (b-c)/b

i.e. iff (a-b)/(b-c) = a/b

Similarly a, b, c are in H.P.

iff 1/b - 1/a = 1/c - 1/b

i.e. iff (a-b)/ab = (b-c)/bc

i.e. iff (a-b)/(b-c) = ab/bc = a/c

### Example

If A, G, H are arithmetic, geometric and harmonic means between two distinct, positive real numbers a and b, show that

- G² = AH i.e. A, G, H are in G.P.
- A, G, H are in descending order of magnitude i.e. A > G > H.

### Solution

Here A = A.M. between a and b = (a+b)/2,

G = G.M. between a and b =
(ab),

H = H.M. between a and b = 2ab/(a+b)

(i) A.H = [(a+b)/2].[2ab/(a+b)] = [(ab)]² = G²

(ii) A - G = (a+b)/2 - (ab) = (1/2)[a +b -2(ab)]
= (1/2)(a -b)²> 0

as (a -b)² is square of a non-zero
real number.

Also G - H = ab - 2ab/(a+b) = [(ab)/(a+b)](a
+ b - 2ab)

= [(ab)/(a+b)](a -b)²> 0

Hence A > G > H.

## Exercise

- Find the n th term of series 5/2 +20/13 + 10/9 + 20/23 +....
- Find the 10th term of series 3 +3/4 +3/7 +....
- If the 10th term of an H.P. is 1/20 and the 17th term is 1/41, find its n th term.
- In an H.P. the p th term is qr and q th term is pr. Show that r th term is pq.
- Find x such that 2 +x, 3 +x, 5 +x may be in H.P.
- Which term of the series 12/17 + 2/3 + 12/19 +.... is 12/25?
- The sum of three numbers in H.P. is 11/12 and sum of their reciprocals is 12. Find the numbers.
- The sum of the reciprocals of three numbers in H.P. is 12 and the product of the numbers is 1/48. Find the numbers.

## Answers

**1.**20/(5n +3)

**2.**3/28

**3.**1/(3n -10)

**5.**1

**6.**9th

**7.**1/2, 1/4, 1/6 or 1/6, 1/4, 1/2

**8.**1/2, 1/4, 1/6 or 1/6, 1/4, 1/2