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**Illustration 1: Let â€˜aâ€™ and â€˜bâ€™ be positive real numbers. If a, A _{1}, A_{2}, b are in arithmetic progression, a, G_{1}, G_{2}, b are in geometric progression and a, H_{1}, H_{2}, b are in harmonic progression, show that**

Hence, A

Now, a, G

So, G

Similarly, as a, H

So, H

And H

Hence, 1/H

So, (H

Now, (G

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

Thus combining the obtained results, we get

(G

**Illustration 2: Let A _{1}, H_{1} denote the arithmetic and harmonic means of two distinct positive numbers. For n â‰¥ 2, let A_{n-1}, H_{n-1 }have arithmetic and harmonic means as An and H_{n }respectively. Then which of the following statements is correct? (2007)**

A

H

H

A

A

Hence, A

A

So, A

â€¦â€¦ â€¦â€¦ â€¦â€¦.

Hence, A

Now, as we have stated above A

So, H

**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.**

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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