Permutations and Combinations

Permutations and Combinations

Question

If A and G are respectively the A.M. and G.M. between the two distinct positive numbers are a and b, then Prove A>G.

Answer

Since A is the A.M. between a and b

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#6 {main}</pre>                                          ...(i)
Since G is the G.M. between a and b

∴                   G = space space square root of ab                                         ...(ii)
A - G =   fraction numerator straight a plus straight b over denominator 2 end fraction minus square root of ab space equals space fraction numerator straight a plus straight b minus 2 square root of ab over denominator 2 end fraction equals space fraction numerator left parenthesis square root of straight a right parenthesis squared plus left parenthesis square root of straight b right parenthesis squared minus 2 square root of straight a square root of straight b over denominator 2 end fraction space equals space fraction numerator left parenthesis square root of straight a minus square root of straight b right parenthesis squared over denominator 2 end fraction greater than 0

∴                          A - G > 0  rightwards double arrow  A > G

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