$\text { now } \quad \frac{n(n+1)}{2}-k-(k+1)=1224 $
$\Rightarrow \quad n^2+n-4 k=2450 $
$\Rightarrow \quad n^2+n-2450=4 k $
$\Rightarrow \quad (n+50)(n-49)=4 k $
$\Rightarrow \quad n>49$
Alternative
$\therefore \quad$ To satisfy this equation $n$ should be of the form of $(4 p+1)$ or $(4 p+2)$ taking $n=50$
$\Rightarrow \quad 4 k =100$
$\Rightarrow \quad k =25$
$\therefore \quad k -20=5$
Now if we take $n=53$
$k =103 $
$n < k$
so not possible
Hence $n \geq 53$ will not be possible.
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