If the sum of the even integers between 1 and k, inclusive, is equal to 2k, what is the value of k?

If is odd, then



while if is even, then the sum would be



The latter case is easier to solve:



which means .

In the odd case, instead of considering the above equation we can consider the partial sums. If is odd, then the sum of the even integers between 1 and would be



Now consider the partial sum up to the second-to-last term,



Subtracting this from the previous partial sum, we have



We're given that the sums must add to , which means




But taking the differences now yields



and there is only one for which ; namely, . However, the sum of the even integers between 1 and 5 is , whereas . So there are no solutions to this over the odd integers.

Idon’t know actually is more difficult