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Solved Examples on Harmonic Progression

Illustration 1: Let ‘a’ and ‘b’ be positive real numbers. If a, A1, A2, b are in arithmetic progression, a, G1, G2, b are in geometric progression and a, H1, H2, b are in harmonic progression, show that

G1G2/ H1H2 = (A1 + A2)/ (H1 + H2) = (2a+b)(a+2b)/9ab. (2002)

Solution: We are given that a, A1, A2, b are in A.P

Hence, A1+A2 = a+b

Now, a, G1, G2, b are in G.P

So, G1G2 = ab

Similarly, as a, H1, H2, b are in H.P

So, H1 = 3ab/ (2b+a)

And H2 = 3ab/ (b+2a)

Hence, 1/H1 + 1/H2 = 1/a + 1/b

So, (H1 + H2)/ H1.H2 = (A1+A2)/G1.G2 = 1/a + 1/b

Now, (G1G2)/ H1.H2 =

= (2a+b) (a+2b)/ 9ab

Thus combining the obtained results, we get

(G1G2)/ H1.H2 = (A1+A2)/(H1 + H2) = (2a+b) (a+2b)/ 9ab

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Illustration 2: Let A1, H1 denote the arithmetic and harmonic means of two distinct positive numbers. For n ≥ 2, let An-1, Hn-1 have arithmetic and harmonic means as An and Hn respectively. Then which of the following statements is correct? (2007)

(a) 1. H1 > H2 > H3 > ….

2. H1 < H2 < H3 < ….

3. H1 > H3 > H5 > …. and H2 < H4 < H6 < ….

4. H1 < H3 < H5 < ….  and H2 > H4 > H6 > ….

(b) 1. A1 > A2 > A3 > ….

2. A1 < A2 < A3 < ….

3. A1 > A3 > A5 > …. and A2 < A4 < A6 < ….

4. A1 < A3 < A5 < ….  and A2 > A4 > A6 > ….

Solution: H1 is the harmonic mean between two numbers ‘a’ and ‘b’.

A1 = (a+b)/2

H1 = 2ab/ (a+b)

Hn = 2An-1H n-1 / (An-1 + H n-1)

An = (An-1 + H n-1)/ 2

A2 is A.M of A1 and H1 and A1 > H1

Hence, A1 > A2 > H1

A3 is A.M of A2 and H2 and A2 > H2

So, A2 > A3 > H2

…… …… …….

Hence, A1 > A2 > A3 > ……

Now, as we have stated above A1 > H2 > H1, A2 > H3 > H2

So, H1 < H2 < H3 < ….

Illustration 3: 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: If a, b and c are in A.P, then

b-a = c-b

This can be written as a, b and c are in A.P. iff b-a = c-b

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

Similarly, when a, b and 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


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