A pack contains `n`cards numbered from 1 to `n`. Two consecutive numbered cards are removed from the pack and the sumof the numbers on the remaining cards is 1224. If the smaller of het numberson the removed cards is `k ,`then `k-20=`____________.
Let two consecutive nymbers are k and `k+1` such that `1leklen-1`, then
`(1+2+3+”…….”+n)-(k+k+1)=1224`
`implies (n(n+1))/(2)-(2k+1)=1224″ or ” k=(n^(2)+n-2450)/(4)`
Now, `1le (n^(2)+n-2450)/(4) le n-1 implies 49ltnlt51`
`:. n =50 implies k=25`
Hence, `k-20=25-20=5`.
Correct Answer – 5
Let number of removed cards be k and `(k+1)`
`:. (n(n +1))/(2) – k – (k +1) = 1224`
`rArr n^(2) +n-4k = 2450 rArr n^(2) + n – 2450 = 4k`
`rArr (n + 50) (n -49) = 4k`
`:. N gt 49`
Let `n = 50`
`:. 100 = 4k`
`rArr k = 25`
Now `k – 20 = 5`