Why the sum of any four consecutive couning numbers is even?

Answer 1

The consecutive numbers are numbers such that n2 - n1 = 1, where the n's are preceding number in the sequence. In other words, n2 follows after n1.

Consider this example:

1, 2, 3 and 4

Then, 1 + 2 + 3 + 4 = 10

How? 1 + 3 is even and 2 + 4 is even. Then, the sum of two even integers is even.

So referring to the example we have, we can see that the sum of two (2k - 1)-th numbers is even. The sum of two 2k-th numbers is even. Combine these sums altogether to obtain the even number.

Make note that the consecutive numbers don't have to start with an odd integer, such as 1. For instance:

Add up 2, 3, 4 and 5, and we obtain even number, which is 14 = 2 + 3 + 4 + 5.

"Symbolically" or simply:

n1 + n3 = 2k where k is any integer

n2 + n4 = 2l where l is any integer

"2k" and "2l" means even sums.

Combine these values altogether to obtain:

(n1 + n3) + (n2 + n4)

= 2k + 2l

= 2(k + l) where k + l belongs to the set of integers

So we obtain even sum of any four consecutive counting numbers.