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For the series $21, 22, 23, ......, k - 1, k;$ the Arithmetic Mean and Geometric Mean of first and last numbers exist inthe given series .If ‘ $K$ ’ is three digit number. Then the number of possible value(s) of ‘ $K$ ’ is/are

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answer is 2.

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Detailed Solution

$A . M = \frac{21 + k}{2}, G . M = \sqrt{21. k} k = 21 λ^{2}, λ \in I a l s o &$

$100 \leq k \leq 999$ and k should be odd
$\frac{100}{21} \leq λ^{2} \leq \frac{999}{21} \Rightarrow 4.76 \leq λ^{2} \leq 47.57$
$λ = 3, 4, 5, 6$ but $λ$ is odd $\Rightarrow 2$

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