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Standard XII

Mathematics

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Let n ∈ N, n>...

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

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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 = $\sqrt{25 n}$ = 5 $\sqrt{n}$ , H = $\frac{50 n}{25 + n}$

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 = $\frac{50 \times 49}{74} = \frac{50 \times 49}{2 \times 37}$ is not a natural number

If n = 81, H = $\frac{50 \times 81}{106}$ is not a natural number

If n = 169, H = $\frac{50 \times 13 \times 13}{38}$ is also not a natural number

If n = 225, H = $\frac{50 \times / 225^{45}}{/ 250^{5}} = 45$ is a natural number

Hence, D is right option.

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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

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