A pack contains $n$ cards numbered from 1 to $n$. Two consecutive numbered cards are removed from th

Question

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 $k-20$ is

Professor PC Thomas Classes · Answer

(5) : We have\ $1+2+3+\ldots+(n-2) \leq 1224 \leq 3+4+\ldots+n$\ $\Rightarrow \frac{(n-2)(n-1)}{2} \leq 1224 \leq \frac{(n-2)}{2}(n+3)$\ Thus $n^{2}-3 n-2446 \leq 0$ and\ $n^{2}+n-2454 \geq 0$\ we get $49<n<51 \quad 	herefore n=50$\ Now, $\frac{n(n+1)}{2}-(2 k+1)=1224 \quad 	herefore k=25$\