Solve this

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 :

 

  1. $(10,97)$

  2. $(11,103)$

  3. $(10,103)$

  4. $(11,97)$


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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