# 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 $7$?

## 4 Answers

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.

• 188250 - 26964 = 161286 @Cm7F7Bb got 149536 – RafaSashi Feb 11 '16 at 13:03
• I've checked mine in python, it's correct. – fosho Feb 11 '16 at 13:06
• I get it! Thanks for answer and explanations. – RafaSashi Feb 11 '16 at 13:08

\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).

• +1 for the correction and progression formula. The acceptance belongs to @Daniel Thanks both! – RafaSashi Feb 11 '16 at 13:10

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

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

• You should at least format your post so that it is more readable, please. – Watson Feb 18 '16 at 12:34
• yes i apologize, it was awful wasnt it. i have reformatted. – Mark Emerson 28DaysIELTS Feb 18 '16 at 13:00