Proving an equation with unorthodox components?

[parabolas]

How do you prove a function when one of the components is only specified as an integer?

For example, how would you prove this function using the (k+1) substitution method?

Prove: n^5 - n is divisible by 10 for all n is greater than or equal to 2.

Thanks in advance.

 

[stapel]

how would you prove this function using the (k+1) substitution method?

What is the definition of "proving" a function?

What is "the (k + 1) substitution method"?

I must confess that I've never heard of either of these. Please provide specifics. Thank you!

Eliz.

 

[Denis]

Hint: if x is an integer and x^5 = k then x and k have same last digit

 

[parabolas]

Proving a function means proving that an equation is true for all values inputed.

Here is the Wikipedia.org article, even though it is a bit dry overly complex.

http://en.wikipedia.org/wiki/Mathematical_proof

The (K + 1) is the primary way I was taught to create a proof. Basically if n = k and you can solve for n = (k + 1), then the equation works for all values of n and the equation is proven.

To be honest, I'm not completely certain as to the concept of proofs either, but that's my best shot.

Hint: if x is an integer and x^5 = k then x and k have same last digit

Thanks for the hint. I deduced something similar to this before, but still am clueless as to how to show this in an acceptable proof form.[/quote]

 

[Deleted member 4993]

The 'k+1' method you are talking about - is formally called 'method of induction.

For that you first show that it is true for the initial number (here it is 2)

n^5 - n is divisible by 10 when n=2.

Assume, it is true that n^5 - n = 10*m

(n+1)^5 - (n+1)

= n^5 + 5n^4 + 10n^3 + 10n^2 + 5n + 1 - n - 1

= (n^5 - n) + 5n(n^3 + 2n^2 + 2n + 1)

= 10m + 5n(n+1)(n^2+n+1)

Now continue....