If the solve the problem
Question:

The sum

$\frac{3 \times 1^{3}}{1^{2}}+\frac{5 \times\left(1^{3}+2^{3}\right)}{1^{2}+2^{2}}+\frac{7 \times\left(1^{3}+2^{3}+3^{3}\right)}{1^{2}+2^{2}+3^{2}}+\ldots \ldots .$

  1. 660

  2. 620

  3. 680 

  4. 600


Correct Option: 1

Solution:

$\mathrm{T}_{\mathrm{n}}=\frac{(3+(\mathrm{n}-1) \times 2)\left(\mathrm{l}^{3}+2^{3}+\ldots+\mathrm{n}^{3}\right)}{\left(1^{2}+2^{2}+\ldots+\mathrm{n}^{2}\right)}$

$=\frac{3}{2} n(n+1)=\frac{n(n+1)(n+2)-(n-1) n(n+1)}{2}$

$\Rightarrow S_{n}=\frac{n(n+1)(n+2)}{2}$

$\Rightarrow S_{10}=660$

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