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 equal to

Answer 1

Let number of removed cards be $k$ and $(k+1)$.
$\therefore \frac{n(n+1)}{2}-k-(k+1)=1224$
$\Rightarrow n^{2}+n-4 k=2450 $
$\Rightarrow n^{2}+n-2450=4 k$
$\Rightarrow (n+50)(n-49)=4 k $
$\therefore n>49 $
Let $ n =50 $
$\therefore 100=4 k \Rightarrow k=25$
Now $ k -20=5$