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Problem

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Prove that

Proof

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For convenience let us call Call the statement

So

This is just a convenient way of referring to it so we don't have to write it over and over


1) For n=1, we have OK


2) We must prove that if is true (for some integer k) then must also be true.

Let us then start with :

We can add the same number to both sides and it will still be true, so let us add 5k+1

To add the terms on the right hand side we will put them over a common denominator:

Taking just the far left and far right parts of the equation:

But this is the statement of ! Since we started with and ended up with without changing the truth value then we have proven 2)

1) and 2) together make the proof for all integers >= 1

QED