Multiples of 3 and 5

We need to find the sum of all positive multiples of $3$ less than $1000$ plus the sum of all positive multiples of $5$ under $1000$ minus the sum of all positive multiples of $3 \times 5 = 15$ under $1000$ , (since we will have counted the multiples of $15$ twice in the first two sums).

As $3 \times 333 = 999$ is the greatest multiple of $3$ less than $1000$ , $5 \times 195 = 995$ is the greatest multiple of $5$ less than $1000$ and $15 \times 66 = 990$ is the greatest multiple of $15$ under $1000$ , the desired sum is

$\dfrac{333}{2}(3 + 999) + \dfrac{195}{2}(5 + 995) + \dfrac{66}{2}(15 + 990) = \boxed{233168}$ .

Note that the sum of an arithmetic sequence of $n$ terms with first term $a_{1}$ and last term $a_{n}$ is $\dfrac{n}{2}(a_{1} + a_{n})$ .