What is the sum of all the natural numbers between $500$ and $1000$.

In general, the sum of an arithmetic progression is $$S_n = \sum_{i=1}^{n} a_i=\frac{n}{2}(a_1+a_n)$$

So, the sum of all even numbers in your interval

$$ = \frac{251}{2}(500+1000)$$

And the sum of all multiples of $14$ in your interval $$ = \frac{36}{2}(504+994)$$

Subtracting these two answers will give you the result you are looking for.

\begin{align*} \sum_{i=0}^{250}(500+2i)-\sum_{i=0}^{35}(504+14i) &=251\cdot500+2\sum_{i=0}^{250}i-36\cdot504-14\sum_{i=0}^{35}i\\ &=251\cdot500+2\biggl[\frac{251(0+250)}2\biggr]-36\cdot504-14\biggl[\frac{36(0+35)}2\biggr]\\ &=161286 \end{align*} using the arithmetic progression formula (see here).