Question

Q.5 If A,G,H are the A.M., G.M. and H.M. of two positive number respectively then prove that ​

Answer 1

Answer:

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Answer 2

Let there are two numbers ‘a’ and ‘b’, a, b > 0

then AM = a+b/2

GM =√ab

HM =2ab/a+b

∴ AM × HM =a+b/2 × 2ab/a+b = ab = (√ab)2 = (GM)2

Note that these means are in G.P.

Hence AM.GM.HM follows the rules of G.P.

i.e. G.M. =√A.M. × H.M.

Now, let us see the difference between AM and GM

AM – GM =a+b/2 – √ab

=(√a2)+(√b)–2√a√b/2

i.e. AM > GM

Similarly,

G.M. – H.M. = √ab –2ab/a+b

=√ab/a+b (√a – √b)2 > 0

So. GM > HM

Combining both results, we get

AM > GM > HM hence proved