Question

Prove that in any nonempty set of n numbers, there is one number whose value is at least the average of the n numbers.

Answer 1

Let us suppose a non empty set S having n elements,

mark the largest among all the elements be M.

and sum of all the numbers Sum =_i S_i  where S_i is element in set at i-th position where i goes from 1 to n.

Also for every i , S_i <= M

for average value should let say N = Sum / n

N = _i S_i / n             

where / stand for divided by

on cross multiplying ,

N*n = _i S_i

as S_i <= M

N*n <=_i M

N*n <= n*M

N <= M

that means N (average) is  always lower than or equal to maximum element in the set S.

Or in other words M is always greater than or equal to average and

hence there exist a element(the largest one in set) in set that is atleast average of set.

PROVED !!

DO UPVOTE IF YOU LIKED MY ANSWER!!