If $1+(1-2^{2} \cdot 1)+(1-4^{2} \cdot 3)+(1-6^{2} \cdot 5)+\ldots+(1-20^{2} \cdot 19) = \alpha - 220 \beta$,then the ordered pair $(\alpha, \beta)$ is equal to:

If the sum of the series $1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + \dots + 2 \cdot (n-1)^2 + n^2$ (when $n$ is odd) is to be determined,given that for even $n$,the sum is $\frac{n(n+1)^2}{2}$,find the sum when $n$ is odd.

$2.\overline{357} = $

If $1+\sin \theta+\sin ^{2} \theta+\ldots \infty = 2 \sqrt{3}+4$,then $\theta = $

For any odd integer $n \ge 1$,${n^3} - {(n - 1)^3} + \dots + {( - 1)^{n - 1}}{1^3} = $

The sum of the series $1 + 3x + 6x^2 + 10x^3 + \dots \infty$ is