Question

The average (arithemetic mean) of the integers from 1 to k, inclusive,
is 651. What is k ?

Answer 1

the sum is k(k+1)/2
so, solve k(k+1)/2 = 651k
k = 1301

or, consider pairs, counting in from the ends.

1+k = 2*651
2+(k-1) = 2*651
...
In each case, k = 1301