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 II by K Sengupta)
The total number of terms is N(given)
Now, the average of the remaining terms after deletion is 271/16 (given)
Then, {sum of (N-1) terms}/(N-1) = 271/16 .....(i)
Since gcd(271,16)=1, it follows that:
n-1=16t=> n=16t+1
Also, from (i), we have:
Sum to (N-1) terms =271*t ......(ii)
Sum to n terms=N(N+1)/2 .......(III)
Let the deleted number be d
Then, d= N(N+1)/2 -271*t
For t=0, we have: N=1
Then, after deletion, the sequence of natural numbers would have 0 terms after deletion. Contradiction.
Now, substituting t=1,2,3,...... we have the following table:
t......N...... Sum to ....... Sum to ..... d ......Remarks
N terms. N-1 terms.
1 17 153 271 -118 X 2 33 561 542 19 v
3 49 1225 813 412 X
4 65 2145 1084 1065 X
Thus, for t>=3, d exceeds n, which is a contradiction as the sequence consists of the first N natural numbers.
Consequently, the required deleted number must be 19.
Edited on February 6, 2022, 11:15 pm
Puzzle Solution: Method III
