Let s _ 1 , s _ 2 , s _ 3 . . . . , s _ 10 respectively be the sum...

Given,

First term of A.P are $1, 2, 3 . . . . 10$, so general term of the first term will be $i$

And common difference are $1, 3, 5, . . . .$, so general term of common difference is given by $2i - 1$

Now sum of the A.P is given by,

${S}_{i} = \dfrac{12}{2}\left[2 \times i + \left(12 - 1\right)\left(2i - 1\right)\right]$

$⇒ {S}_{i} = 6\left[2 \times i + 11\left(2i - 1\right)\right]$

$⇒ {S}_{i} = 144i - 66$

So, $\mathop{∑}_{i = 1}^{10}{S}_{i} = \mathop{∑}_{i = 1}^{10}144i - 66\mathop{∑}_{i = 1}^{10}1$

$⇒ \mathop{∑}_{i = 1}^{10}{S}_{i} = 144\left(\dfrac{10 \times 11}{2}\right) - 66 \times 10$

$⇒ \mathop{∑}_{i = 1}^{10}{S}_{i} = 792\left(10\right) - 66 \times 10$

$⇒ \mathop{∑}_{i = 1}^{10}{S}_{i} = 7260$

Hence this is the correct option.