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

Answer: 5

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