# Linked Questions

5 votes
1 answer
441 views

• 2,538
2 votes
4 answers
144 views

### A more rigorous way to prove this? [duplicate]

I would like to prove the following statement $$x^n-a^n=(x-a)\sum^{n-1}_{k=0}x^ka^{n-k-1},\qquad\forall n\in\Bbb N_0$$ I can easily prove it by induction using polynomial long division or series ...
• 5,626
1 vote
2 answers
7k views

### Prove that $5^n - 2^n$ is divisible by $3$ for all nonnegative integers $n$ using mathematical induction [duplicate]

Using mathematical induction, prove for all integers n 1 that $5^n - 2^n$ is divisible by 3. Can someone help me with this?
• 643
1 vote
3 answers
121 views

### How to prove $z^n - z^n_0 = (z-z_0) \sum_0^{n-1} z^kz_0^{n-1-k}$ [duplicate]

I want to prove that with $z_0$ a root of $1+z^n$, I have $$z^n - z^n_0 = (z-z_0)\sum_0^{n-1} z^kz_0^{n-1-k}$$
• 393
0 votes
3 answers
111 views

### How can I use induction solve this?

How can I show/solve this? I've tried by using the basis step and the inductive step, but just can't seem to get it right. $$\forall(n \geq 0)(4\mid(9^n − 5^n)).$$
• 19
0 votes
3 answers
95 views