If $2^{3} + 4^{3} + - - - - + (2 n)^{3} = k n^{2} (n + 1)^{2},$ then $k =$

\frac{1}{2}

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1

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\frac{3}{2}

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2

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Solution

The correct option is D $2$
Given,
$2^{3} + 4^{3} + \dots + (2 n)^{3} = k n^{2} (n + 1)^{2},$
or, $2^{3} + (2.2)^{3} + \dots + (2. n)^{3} = k n^{2} (n + 1)^{2}$
or, $2^{3} (1^{3} + (2)^{3} + \dots + (n)^{3}) = k n^{2} (n + 1)^{2}$
or, $2^{3} . {(\frac{n (n + 1)}{2})}^{2} = k n^{2} (n + 1)^{2}$
or, $\frac{8}{4} = k$
or, $k = 2$ .

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