How to solve this problem, I can not figure it out:

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.

Find the sum of all the multiples of 3 or 5 below 1000.

The previously posted answer isn't correct. The statement of the problem is to sum the multiples of 3 and 5 below 1000, not up to and equal 1000. The correct answer is
\begin{eqnarray}
\sum_{k_{1} = 1}^{333} 3k_{1} + \sum_{k_{2} = 1}^{199} 5 k_{2} - \sum_{k_{3} =1}^{66} 15 k_{3} = 166833 + 99500 - 33165 = 233168,
\end{eqnarray}
where we have the used the identity
\begin{eqnarray}
\sum_{k = 1}^{n} k = \tfrac{1}{2} n(n+1).
\end{eqnarray}