What is the sum of all the natural numbers between $500$ and $1000$. What is the sum of all the natural numbers between $500$ and $1000$ (extremes included) that are multiples of $2$ but not of $7$?
 A: 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.
A: \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).
A: hint1: for divisibility by 2 find the sum of arithmetic numbers of distance2
hint2: Numbers divisible of 7 are either multiples of 14 or divisible by 14 when added 7, so try to subtract sequence of difference 14 from your sum
A: Heres a code solution using a C# console application:

// hold all numbers to be summed 

List<int> multiplesOfTwoCollection = new List<int>();<br>

// count numbers 

for (int naturalNumber = 500; naturalNumber <= 1000; naturalNumber++)

{ if (naturalNumber % 2 == 0) { multiplesOfTwoCollection.Add(naturalNumber); }

// log contents

 Console.WriteLine("natural number:{0}", naturalNumber); }

// multiples of 7

int sumOfInts = 0;
foreach (var item in multiplesOfTwoCollection)
 if (item % 7 == 0) { Console.WriteLine("multiple of 7:{0}", item);
     } else { sumOfInts += item; }

Console.WriteLine("sum of numbers between 500 and 1000 that are multiples of 2 not 7:{0}", sumOfInts);

Returns
161286
