1. ## Proof

I am studying Real analysis , and there is a question that I am not sure if I should ask it under Number Theory section but I will ask it anyway
Does anyone know how to prove this?

an - bn = (a-b) (an-1 + an-2b +....... + abn-2 + bn-1)

Does anyone know how to prove this or know at least to which category does this kind of question goes to so I can ask to the right place? Thank you.

2. ## Re: Proof

Originally Posted by davidciprut
I am studying Real analysis , and there is a question that I am not sure if I should ask it under Number Theory section but I will ask it anyway
Does anyone know how to prove this?

an - bn = (a-b) (an-1 + an-2b +....... + abn-2 + bn-1)

Does anyone know how to prove this or know at least to which category does this kind of question goes to so I can ask to the right place? Thank you.
I don't know what you mean by "proof" here...just multiply it out:
$\displaystyle (a - b)(a^{n-1} + a^{n - 2}b + a^{n - 3}b^2 + \text{ ... } + a b^{n - 2} + b^{n - 1})$

$\displaystyle = (a^n + a^{n - 1}b + a^{n - 2}b^2 + \text{ ... } + a^2 b^{n - 2} + a b^{n - 1})$ $\displaystyle - (a^{n - 1}b + a^{n - 2} b^2 + \text{ ... } + ab^{n - 1} + b^n)$

Now subtract.

(And yes, this would be better placed in the Algebra forum. Don't sweat it.)

-Dan

3. ## Re: Proof

Thanks for helping out with the question and correcting me with subject. I am studying first year math and we are studying Infinitesimal Calculus so it's kinda hard to tell what we are doing sometimes.

Some people told me to solve it by induction but it didn't work. Anyway thank you and appreciate the help!

4. ## Re: Proof

Originally Posted by davidciprut
Thanks for helping out with the question and correcting me with subject. I am studying first year math and we are studying Infinitesimal Calculus so it's kinda hard to tell what we are doing sometimes.!
I have a question for you. By Infinitesimal Calculus do you mean non-standard analysis?
If so, what does that simple algebra question have to do with that?

Here is some free material on that:
In Elementary Calculus: An Infinitesimal Approach
In Elementary Caculus: An Infinitesimal Approach by Jerome Keisler The chapters and whole book is a free down-load at

5. ## Re: Proof

That's actually a good questions. The lectures has nothing to do with this... This is a question from my homework that was given from the exercise. It has nothing to do with the lecture. The lecture is much harder. Our homework is more technical when it comes to proving thins like the question I asked.

I can only tell you what we are studying because I don't know what is a non standard analysis is. It's my first year in the University and it's kinda a new to me these terms... (infinitesimal calculus etc.)

We first learned fields and its axioms, and some basic things with proofs like why 0.a=a and things like that.
Now we are learned Supremum and infimum with proof and its really hard and we are going to learn the whole calculus (limits derivatives and so on) with proofs. If there are more books that you guys recommend I am open to suggestion because it's going really hard.