Mathematics - Sequence & Series Question with Solution | TestHub

Clearly,   1 + 2 + 3+.......+n - 2$\le$1224$\le$3 + 4+.......n
$\Rightarrow \frac{\left(\text{n}-2\right)\left(\text{n}-1\right)}{2}\le 1224\le \frac{\left(\text{n}-2\right)}{2}\left(3+\text{n}\right)$
$\Rightarrow {\text{n}}^2-3\text{n}-2446\le 0\text{ and }{\text{n}}^2+\text{n}-2454\ge 0$
$\Rightarrow 49<\text{n}<5\mathrm{1 }\Rightarrow \text{n}=50$
$\therefore \frac{\text{n}\left(\text{n}+1\right)}{2}-\left(2\text{k}+1\right)=122\mathrm{4 }\Rightarrow \text{k}=25\Rightarrow \text{k}-20=5$