Mathematics - Sequence & Series Question with Solution | TestHub

$\mathrm{S}=1+\sum _{\mathrm{r}=1}^{10}1-{\left(2\mathrm{r}\right)}^2\left(2\mathrm{r}-1\right)$

$=1+10-\sum _{\mathrm{r}=1}^{10}\left(8{\mathrm{r}}^3-4{\mathrm{r}}^2\right)$

$=11-\left[8{\left(\frac{10\times 11}{2}\right)}^2-4\left(\frac{10\times 11\times 21}{6}\right)\right]$

$=11-\left[2{\left(110\right)}^2-140\times 11\right]$

$=11-22\left(1100-70\right)$

$=11-220\left(110-7\right)$

$\therefore$ $\alpha -220\beta$$=11-220\left(103\right)$

$\therefore$ $\alpha =11,\beta =103$

$\left(\alpha \mathit{,}\beta \right)=\left(11,103\right)$.