The sum frac{3 times 1}{1^2} + frac{5 times ( | vedclass

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(B) The $n^{th}$ term $T_n$ is given by:$T_n = \frac{(2n+1) \sum_{k=1}^n k^3}{\sum_{k=1}^n k^2}$Using the formulas $\sum k^3 = \frac{n^2(n+1)^2}{4}$ a

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$660$

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(B) The $n^{th}$ term $T_n$ is given by:$T_n = \frac{(2n+1) \sum_{k=1}^n k^3}{\sum_{k=1}^n k^2}$Using the formulas $\sum k^3 = \frac{n^2(n+1)^2}{4}$ and $\sum k^2 = \frac{n(n+1)(2n+1)}{6}$:$T_n = \frac{(2n+1) 	imes \frac{n^2(n+1)^2}{4}}{\frac{n(n+1)(2n+1)}{6}}$$T_n = \frac{6}{4} n(n+1) = \frac{3}{2}(n^2 + n)$Now,find the sum $S_{10} = \sum_{n=1}^{10} T_n = \frac{3}{2} \left[ \sum_{n=1}^{10} n^2 + \sum_{n=1}^{10} n ight]$$S_{10} = \frac{3}{2} \left[ \frac{10(11)(21)}{6} + \frac{10(11)}{2} ight]$$S_{10} = \frac{3}{2} [385 + 55] = \frac{3}{2} [440] = 660$

The sum $\frac{3 \times 1}{1^2} + \frac{5 \times (1^3 + 2^3)}{1^2 + 2^2} + \frac{7 \times (1^3 + 2^3 + 3^3)}{1^2 + 2^2 + 3^2} + \dots$ up to the $10^{th}$ term is:

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