I want to have a little 'le cry'.

Earlier today, I made a super detailed and tidy post on my question, but for some reason it didn't post(I think it was because of my internet), and now I've got to type it all again, and with less care and attention to detail.

Anyways, not so long ago, I had a query on this site regarding Inductive proofs, but now I've reached a point in the questions that I don't really see where I am meant to go in the inductive step.

Just to clarify, from what I have learnt, Inductive Proofs are:

Proving a statement is true for the defined values of a variable(e.g all positive integers) by

taking the next step of one side of the statement and adding it to the other, and then using

simplifying and factorising techniques to change every variable on the other side to the variable + 1.

Questions that seem relatively straightforward to me that I have been mowing through include:

Prove by induction that:

1 + 3 + 5 + 7 + ... + (2n-1) = n^{2 }where n ≥ 1.

These are easy, as the next step on the left hand side is (2(n+1)-1) which is added to both sides and factorise to produce the right hand side as (n+1)^{2}

But now I have reached questions that have full on stopped me in my tracks like:

Question 1Which has 2 variables?!)

Use proof by induction to prove that:

(x-1) is a factor of x^{n}-1

when n ≥ 0.

Question 2Inequalities?)

Use proof by induction to prove that:

1 x 2 x 3 x 4 x 5 x 6 x ... x n ≥ 3^{n }When n > 6.

Question 3I would know how to do this kind of question in a proof by exhaustion or something)

Use proof by induction to prove that:

7^{n} + 2 x 13^{n}

is a multiple of 3 when n ≥ 0.

Which all have me soooo confused. Is it my understanding of how proof by induction works that is making me go off?

It would be more possible for me to think around it if it wasn't a proof by induction, but these have me having no clear idea of where to go.

I do have some working out, but they are definitely not worth any attention for now at least.

Please don't give me the answers, since I have a limited number of these questions, but maybe a pointer in the right direction or something would be great!

Thanks!