A pack contains $n$ cards numbered from $1$ to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is $k$, then $k - 20 =$

$\text{Total sum of all cards } = 1 + 2 + \cdots + n = \dfrac{n(n+1)}{2}$ $\text{Removed cards } = k \text{ and } k+1$ $\text{Remaining sum } = 1224$ $\Rightarrow \dfrac{n(n+1)}{2} - (2k+1) = 1224$ $\text{To find: } k - 20$ $1.\ S_n = \dfrac{n(n+1)}{2}$ $2.\ S_n - (2k+1) = 1224$ $3.\ \text{find } n \Rightarrow k \Rightarrow k-20$ $k \ge 1 \Rightarrow 2k+1 \ge 3$ $S_n = 1224 + (2k+1) \ge 1227$ $S_{48} = \dfrac{48 \times 49}{2} = 1176$ $S_{49} = \dfrac{49 \times 50}{2} = 1225$ $1225 < 1227 \Rightarrow \text{not possible}$ $S_{50} = \dfrac{50 \times 51}{2} = 1275$ $S_{51} = \dfrac{51 \times 52}{2} = 1326$ $S_{51} - 1224 = 102\ (\text{even}) \Rightarrow \text{no integer } k$ $2k+1 = 1275 - 1224 = 51$ $2k = 50$ $k = 25$ $k+1 = 26 \le 50\ \checkmark$ $1275 - (25+26) = 1224\ \checkmark$ $\boxed{k - 20 = 25 - 20 = 5}$