A professor writes N consecutive natural numbers, beginning with 1, on the blackboard. One of the students in the class deletes one of the numbers (exactly one number), from that list.
Now, given that the average of the remaining N-1 numbers is 271/16.
Can you find out the number that has been deleted from the list ?
(In reply to
Puzzle Solution: Method I by K Sengupta)
Average of the terms with the exclusion of m
= {N(N+1)/2 - m}/(N-1)
= {N(N+1)-2m}/{2(N-1)}
Since by the problem, this average is 271/16, then we must have:
{N(N+1)-2m}/{2(N-1)} =271/16
=> 8{N(N+1) - 2m} =271/(N-1)
=> 16m = 8N(N+1) - 271(N-1)
=> m = N(N+1)/2 - {271(N-1)/16} .....(i)
Then N-1 must be a multiple of 16.
So, N= 1,17,33,49,65,81, an so on....
If N=1, then from (I), we obtain m=1, which is untenable.
If N= 17 => m= 17*9-271= -138<0
If N=33 => m-N>=763
If N=49 => m-N >= 1061
Accordingly, for N >=49, the value of m will always exceed N.
This is a contradiction.
Therefore, only (N,m)=(33,19) satisfies all the given conditions, and:
Consequently, the required deleted number is 19
Edited on January 27, 2022, 2:04 am
Puzzle Solution: Method II
