Linked Questions

2 votes
4 answers
146 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 ...
Ali Caglayan's user avatar
  • 5,756
1 vote
3 answers
132 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 $$ z^n - z^n_0 = (z-z_0)\sum_0^{n-1} z^kz_0^{n-1-k}$$
dcholleton's user avatar
0 votes
3 answers
124 views

Prove by induction: $b^n-a^n=(b-a)\cdot \sum_{k=0}^{n-1}{a^k\cdot b^{n-1-k}}$ $\forall n\in \mathbb{N}$ and $a,b\in \mathbb{R}$ [duplicate]

Prove by induction: $b^n-a^n=(b-a)\cdot \sum_{k=0}^{n-1}{a^k\cdot b^{n-1-k}}$ $\forall n\in \mathbb{N}$ and $a,b\in \mathbb{R}$ $\exists n\in \mathbb{N}:b^n-a^n=(b-a)\cdot \sum_{k=0}^{n-1}{a^k\cdot b^...
user avatar
0 votes
2 answers
189 views

Induction Proof that $ x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$ [duplicate]

I seek an inductive proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\cdots+xy^{n-2}+y^{n-1}).$ I am stuck.
Gustavo Ortega's user avatar
1 vote
2 answers
8k 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?
1ftw1's user avatar
  • 643
2 votes
5 answers
6k views

How do I prove that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\dotsb+xy^{n-2}+y^{n-1})$

I am unsuccessfully attempting a problem from Spivak's popular book 'Calculus' 3rd edition. The problem requires proof for the following equation: $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}+\dotsb+xy^{n-2}+y^{n-...
od320's user avatar
  • 23
2 votes
1 answer
4k views

The identity $a^n-b^n=(a-b) (\sum_{i=0}^{n-1}a^ib^{n-1-i})$ [duplicate]

How do I use finite induction to prove that $$a^n-b^n=(a-b) (\sum_{i=0}^{n-1}a^ib^{n-1-i}), \forall a,b\in \Bbb{R}\space \text{and} \space \forall n \in \Bbb{N}?$$ Ok, for $n=2$ it's fine. $a^2-b^2=(a-...
Derso's user avatar
  • 2,773
4 votes
2 answers
1k views

Is there a name for a binomial expansion without coefficients?

I am investigating a problem from George E. Andrews Number Theory (Dover, 1971), discussed previously here: Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$ I was led ...
billc's user avatar
  • 168
4 votes
1 answer
519 views

Decomposition of $x^n-y^n$

As a part of textbook assignment I was asked to prove that $x^2-y^2=(x+y)(x-y)$, and I did so as follows: $$x^2-y^2=x^2-y^2+xy-xy=x(x+y)-y(x+y)=(x+y)(x-y)$$ Later, I used similar method to decompose $...
Misha.P's user avatar
  • 211
3 votes
6 answers
185 views

Proof that: $(a-b)\cdot\Bigg(\sum_{k=0}^{n}a^{n-k}b^{k}\Bigg)=a^{n+1}-b^{n+1}\text{ }\forall n\in\mathbb{N_{0}}$

I'm trying to prove a more general version of the 3rd binomial equation via mathematical induction which will help me complete another proof. $$(a-b)\cdot\Bigg(\sum_{k=0}^{n}a^{n-k}b^{k}\Bigg)=a^{n+1}...
Christian Singer's user avatar
0 votes
3 answers
115 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. $$\forall(n \geq 0)(4\mid(9^n − 5^n)).$$
Maren's user avatar
  • 17
2 votes
1 answer
57 views

For $a \gt b \gt 0$ prove that $a^n – b^n \geq n(a-b) (ab)^{\frac{n-1}{2}}$

For $a \gt b \gt 0$ prove that $a^n – b^n \geq n(a-b) (ab)^{\frac{n-1}{2}}$ What is an appropriate way to solve this?
Kundu Sandip's user avatar
5 votes
0 answers
119 views

Verify proof: $a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$

A short disclaimer: I do know this question has been asked multiple times here and several answers (including combinatorics) have been given already. However, among all these posts, I did not find ...
kaithkolesidou's user avatar
0 votes
1 answer
79 views

Proof by induction valid or not?

Prove by induction the following: $$\sum_{i=0}^n x^i = \frac{1-x^{n+1}}{1-x}$$ We want: $$x^0+x^1+ \ldots + x^n = \frac{1-x^{n+1}}{1-x}$$ I try this for $i=1$ and it works, so I have an initial ...
user3200098's user avatar
  • 1,227
0 votes
1 answer
114 views

$x^n-y^n$ equivalence

I have questions about a formula that is : $x^n - y^n = (x-y)(x^{n-1} + x^{n-2}y + \ldots + xy^{n-2} + y^{n-1})$ That's how it's written on my textbook but it seems like I've trouble understanding it ...
user15757055's user avatar