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Permutations And Combinations

Question
CBSEENMA11013124

If two G.M.'s WiredFaculty and one A.M. 'A' be inserted between two numbers, show that

2 straight A space equals space fraction numerator straight g subscript 1 superscript 2 over denominator straight g subscript 2 end fraction space plus space fraction numerator straight g subscript 2 superscript 2 over denominator straight g subscript 1 end fraction

Solution

Let a and b be two numbes, since g1 and g2 are two G.M.'s between a and b.

∴  a, straight g subscript 1 comma space space straight g subscript 2, b are in G.P.
Let r be common ratio

∴                    straight g subscript 1 space equals space ar comma space space straight g subscript 2 space equals space ar squared comma space straight b space equals space ar cubed
Since A is the A.M. between a and b

∴                      straight A space equals space fraction numerator straight a plus straight b over denominator 2 end fraction
rightwards double arrow                   2A = a + b                                                                ...(i)
                   R.H.S. = fraction numerator straight g subscript 1 superscript 2 over denominator straight g subscript 2 end fraction plus fraction numerator straight g subscript 2 superscript 2 over denominator straight g subscript 1 end fraction space equals space fraction numerator straight a squared straight r squared over denominator ar squared end fraction plus fraction numerator straight a squared straight r to the power of 4 over denominator ar end fraction space equals space straight a plus ar cubed space equals space straight a plus straight b space equals space 2 straight A  {By using (i)}
                             = L.H.S.

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