Question

Mr. Laschober's Claim

"I believe that there exists integers
that cannot be expressed as a
sum of consecutive integers."
- Mr. Laschober
How can I prove this wrong?

Answer 1

Ok, if we start with a number n, then we can write the consecutive numbers of n as:

n + 1

n + 2

n + 3..

etc.

Now, suppose that we have an integer x, and we want to write it as a sum of consecutive numbers.

There is a really trivial way.

We know that x + 0 = x.

now, we can write:

0 = (x - 1) + (x - 2) + ... + 1 + 0 + (-1) + (-2) + ..... + (-x + 2) + (x - 1)

Then we wrote zero as a sum of consecutive numbers.

Now we can write:

x = x + 0 = x + (x - 1) + (x - 2) + .... + 1 + 0 - 1 - 2 - ... - (x - 1) = x

Then we show that for any integer x, we can write it as a sum of consecutive integers.

Then it must work for all the integers x, then we prove that all the integers can be written as a sum of consecutive integers.