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Standard XII

Mathematics

AM,GM,HM Inequality

Let n∈ N,n>...

Question

Let $n \in N, n > 25$ .Let A,G,H denote arithmetic mean, geometric mean and harmonic mean respectively of $25$ and $n$ .The least value of $n$ for which $A, G, H \in {25, 26, . . ., n}$ is ?

49

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81

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169

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225

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Solution

The correct option is D $225$
$A = \frac{25 + n}{2}, G = 5 \sqrt{n}, H = \frac{50 n}{25 + n}$
As A, G, H are natural numbers, n must be odd perfect square. Now, H will be natural number, if we take $n = 225.$

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

The correct option is D 225A=25+n2, G=5√n,H=50n25+nAs A, G, H are natural numbers, n must be odd perfect square. Now, H will be natural number, if we take n=225.

Let $n \in N, n > 25$ .Let A,G,H denote arithmetic mean, geometric mean and harmonic mean respectively of $25$ and $n$ .The least value of $n$ for which $A, G, H \in {25, 26, . . ., n}$ is ?

The correct option is D $225$
$A = \frac{25 + n}{2}, G = 5 \sqrt{n}, H = \frac{50 n}{25 + n}$
As A, G, H are natural numbers, n must be odd perfect square. Now, H will be natural number, if we take $n = 225.$