The sum of all numbers from 8 to 18 is an even number?

Answer 1

If you include 8 and 18 that is 11 numbers and their sum is 143 which is odd ---- Explanations (by the way, it is really not all the numbers between 8 and 18, it is just all the integers. There is an infinite number of numbers between 8 and 18, think of just 8, 8.1, 8.11, 8.111, 8.1111 etc)

Here are some other fun ways to do this:

One way to do it, of course is just to add

8+

9

10

11

12

13

14

15

16

17

18

Before you add them all, consider just adding the last column, and looking at the ones column of that number.

8+9+0+1+2+....+8 that number is 53 so the sum is odd.

If that numbers is even then the number is even

OR Looking at just the odd numbers. The sum of even numbers is always even. An odd number and an even number is an odd number. So if the sum of the odd numbers is even then the last column will be an odd number and the total sum will be odd. The sum of odd numbers is even when there is an even number of them. Look at 2 odd numbers.

(ie 3+3 =6, this is two odd numbers and their sum is even)

9,11,13,15,17 are even and there are 5 of them. This is an odd number of odd so the sum is odd. Now we know the sum of the odds and the evens is odd so the total of the last column must be odd and therefore the sum is odd.

Another way is to see that the sum of the first n numbers is n(n+1)/2

We can add the sum of the first 18 numbers then subtract the sum of the first 7 numbers and what is left is the sum of the numbers 8 to 18.

So in this case we have 18x19/2-8x9/2 which is

(18x19-7x8)/2 this simplifies by first dividing the 18 and the 8 by 2 so we have

9x19-4x7=

171-28=143

OR What if we match up and add the 8 and the 18, the 9 and the 17, the 10 and the 16, the 11 and the 15, the 12 and the 14 and you have 13 left. This is the idea behind the formula for the first n numbers. That is 26x5+13=143