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

You can make a start by putting together an expression for the sum of the even integers between 1 and k inclusive.

Let S be the sum of the even integers between 1 and k inclusive.

Then:

S=2+4+6+⋯ + (k-2) + k

As k is even, you can say r = 2k and so:

S=2 (1+2+3+⋯ + (r-1) + r)

Now the sum of the first r numbers is well-known, it's the r th triangle number and we have:

1+2+3+⋯ + (r-1) + r = r (r+1) / 2

Now we can keep it simple and say 2k=4r and so:

S=2 (1+2+3+⋯ + (r-1) + r) = 4r=2 r (r+1) 2 = r (r+1)

So you can build a quadratic in r and so get k.