(D) The given series is $S_n = 2^3 + 4^3 + 6^3 + \ldots + (2n)^3$. We can write this as $S_n = \sum_{k=1}^{n} (2k)^3$. $S_n = \sum_{k=1}^{n} 8k^3 = 8

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(D) The given series is $S_n = 2^3 + 4^3 + 6^3 + \ldots + (2n)^3$. We can write this as $S_n = \sum_{k=1}^{n} (2k)^3$. $S_n = \sum_{k=1}^{n} 8k^3 = 8 \sum_{k=1}^{n} k^3$. Using the formula for the sum of cubes of the first $n$ natural numbers,$\sum_{k=1}^{n} k^3 = [\frac{n(n+1)}{2}]^2$. So,$S_n = 8 \times [\frac{n(n+1)}{2}]^2 = 8 \times \frac{n^2(n+1)^2}{4} = 2n^2(n+1)^2$. Comparing this with the given expression $h n^2(n+1)^2$,we get $h = 2$.

Class 11 Mathematics Sequences and Series nth term of special series, Sum to n terms and Infinite number of terms

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If $2^3+4^3+6^3+\ldots+(2n)^3=h n^2(n+1)^2$,then $h$ is equal to

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A
$1/2$
B
$1$
C
$3/2$
D
$2$

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nth term of special series, Sum to n terms and Infinite number of terms

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