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Question

Let n ϵ N, n > 25. Let A, G, H denote arithmetic mean, geometric mean and harmonic mean of 25 and n. The least value of 'n' for which A, G, H ϵ {25, 26, - - - -, n} is


A

49

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B

81

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C

169

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D

225

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Solution

The correct option is C

225


A = 25+n2, G = 25n = 5 n, H = 50n25+n

For A to be natural number, n should be an odd number

For G to be natural number, n has to be a perfect square

Hence, n should be both odd number & perfect square.

If n = 45, H = 50×4974=50×492×37 is not a natural number

If n = 81, H = 50×81106 is not a natural number

If n = 169, H = 50×13×1338 is also not a natural number

If n = 225, H = 50×/22545/2505=45 is a natural number

Hence, D is right option.


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