Question:
If
$1+\left(1-2^{2} .1\right)+\left(1-4^{2} .3\right)+\left(1-6^{2} .5\right)+\ldots . .+\left(1-20^{2} .19\right)$
$=\alpha-220 \beta$, then an ordered pair $(\alpha, \beta)$ is equal
to :
Correct Option: , 2
Solution:
$1+\left(1-2^{2} .1\right)+\left(1-4^{2} .3\right)+\ldots \ldots+\left(1-20^{2} .19\right)$
$=\alpha-220 \beta$
$=11-\left(2^{2} .1+4^{2} .3+\ldots \ldots .+20^{2} .19\right)$
$=11-2^{2} \cdot \sum_{\mathrm{r}=1}^{10} \mathrm{r}^{2}(2 \mathrm{r}-1)=11-4\left(\frac{110^{2}}{2}-35 \times 11\right)$
$=11-220(103)$
$\Rightarrow \alpha=11, \beta=103$
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