Prove $ X^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1) - Math Stack Exchange

1. 1. Home
   2. Questions
   3. Tags
   4. Users
   5. Unanswered
2. Teams

   Ask questions, find answers and collaborate at work with Stack Overflow for Teams.

   Try Teams for free Explore Teams
3. Teams

4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams

Teams

Q&A for work

Connect and share knowledge within a single location that is structured and easy to search.

Learn more about Teams Prove $ x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$ Ask Question Asked 10 years, 3 months ago Modified 5 months ago Viewed 91k times 11 $\begingroup$

So what I am trying to prove is for any real number x and natural number n, prove $$x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$$

I think that to prove this I should use induction, however I am a bit stuck with how to implement my induction hypothesis. My base case is when $n=2$ we have on the left side of the equation $x^2-1$ and on the right side: $(x-1)(x+1)$ which when distributed is $x^2-1$. So my base case holds.

Now I assume that $x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$ for some $n$. However, this is where I am stuck. Am I trying to show $x^{n+1}-1=(x-1)(x^n + x^{n-1}+x^{n-2}+...+x+1)$? I am still a novice when it comes to these induction proofs. Thanks

• elementary-number-theory
• proof-writing

Share Cite Follow edited Aug 17, 2014 at 9:55 Petite Etincelle 14.9k22 gold badges3333 silver badges6161 bronze badges asked Aug 17, 2014 at 9:52 chettylicechettylice 11311 gold badge11 silver badge44 bronze badges $\endgroup$ 4

• $\begingroup$ proofwiki.org/wiki/Sum_of_Geometric_Progression $\endgroup$ – lab bhattacharjee Commented Aug 17, 2014 at 9:53
• $\begingroup$ You don't need induction. Just repeat the proof as in the case $n=2$. $\endgroup$ – Quang Hoang Commented Aug 17, 2014 at 9:53
• $\begingroup$ Prove it for n=1; then, assuming that it is true for n=k, try to show that it is true for n=k+1. It is easy indeed. (I do not know the name of this proving method.) $\endgroup$ – enthu Commented Aug 17, 2014 at 10:11
• $\begingroup$ @EnthusiasticEngineer It is called Proof by Induction $\endgroup$ – BadAtAlgebra Commented Feb 17, 2019 at 7:02

Add a comment |

2 Answers 2

Sorted by: Reset to default Highest score (default) Date modified (newest first) Date created (oldest first) 15 $\begingroup$

To conclude your induction proof, just multiply x both sides :

$x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1) $

multiply $x$ both sides :

$\begin{align} \\ x^{n+1}-x &=(x-1)(x^n+x^{n-1}+x^{n-2}+...+x^2+x) \\ x^{n+1}-1 -(x-1) &=(x-1)(x^n+x^{n-1}+x^{n-2}+...+x^2+x) \\ x^{n+1}-1 &=(x-1)(x^n+x^{n-1}+x^{n-2}+...+x^2+x)+(x-1) \\ \end{align}$

factor $(x-1)$ and you're done !

Share Cite Follow edited Aug 17, 2014 at 10:11 answered Aug 17, 2014 at 10:03 AgentSAgentS 12.2k33 gold badges3838 silver badges7474 bronze badges $\endgroup$ 2

• 1 $\begingroup$ Your sums should end in $\dots+x^2+x$, not $\dots+x+x$. $\endgroup$ – Joonas Ilmavirta Commented Aug 17, 2014 at 10:10
• $\begingroup$ Ahh yes ! will fix it thanks :) $\endgroup$ – AgentS Commented Aug 17, 2014 at 10:11

Add a comment | 0 $\begingroup$

You may just open hooks in right half of expression: $$(x - 1)(x^{n - 1} + x^{n - 2} + \dots + x + 1) = x * (x^{n - 1} + x^{n - 2} + \dots + x + 1) - 1 * (x^{n - 1} + x^{n - 2} + \dots + x + 1) = (x^n + x^{n - 1} + \dots + x^2 + x) - (x^{n - 1} + x^{n - 2} + \dots + x + 1) = x^ n - 1 $$ No problems)

Share Cite Follow answered Apr 28, 2019 at 9:05 Алексей КирилловАлексей Кириллов 1 $\endgroup$ Add a comment |

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged

• elementary-number-theory
• proof-writing

.

• Featured on Meta
• We’re (finally!) going to the cloud!
• More network sites to see advertising test [updated with phase 2]

Linked

0 Prove that $1+a+a^2+...+a^n$ = $\frac{1-a^{n+1}}{1-a}$ for all $a\ne 1$. 0 How can we prove that x^n - 1 always be a multiple of x - 1 1 Solutions of $1+x+x^2+...+x^n$

Related

2 Divisibility problem involving the floor function 3 Prove by induction that $10^n -1$ is divisible by 11 for every even natural number 0 Proof by induction with a prime number 1 Prove by careful induction that if $r \in \mathbb{N}$, then $b^{2^r} - 1 = (b-1)(b+1)(b^2 + 1)(b^{2^{2}} + 1) ... (b^{2^{r-1}} + 1)$ 1 Prove $x>nδ_n$, $\forall n∈\mathbb{N}+$ for $δ_n := \sqrt[n]{x}−1$ 1 Alternative Induction Format 1 Am I permitted to use the truth of the base case during the inductive step in a proof using weak induction? 3 Prove that he can bring back this amount of coins 0 Prove by induction that for a graph $G=(V, E)$, if it doesn't contain a cycle, then $|E|\leq|V|-1$.

Hot Network Questions

• Can not update chrome in ubuntu
• What are Christian responses to Carlo Alvaro's argument against Christian theism?
• Why is the term "card" used in "expansion card"?
• To which extent I should let my PI know that I am not feeling very well with my PhD study
• How can I use a terminal emulator as an efficient application launcher?
• Could rocket exhaust eventually lead to detrimental effects from interplanetary space pollution?
• Can a thunderstorm affect a satellite in low earth orbit?
• How a person become a rabbi around 1 CE?
• Where can I find an up-to-date map of Stockholm Central Station that shows the platform layout?
• Can you omit から directly connecting plain forms?
• How to demystify why my degree took so long on my CV
• How to test a programmer's ability to handle a large code base?
• Pros & cons of Sallen-Key vs. double RC cascaded stages LPF topologies
• How to make seal on lever flip-top "Kilner" jars without spoiling sterilization?
• How does the Born rule arise in the many-worlds interpretation with only two worlds?
• Most Efficient Glide: Pitch Up or Level Flight to Bleed Airspeed
• Use of “12 m.” for noon and “12 p.m.” for midnight
• Alexander's dictum
• PSE Advent Calendar 2024 (Day 1): A Snowy Christmas
• Is it possible to use NAS hard drives in a desktop?
• I have a Greek seasonal residence permit. Can I go back to my country from Poland instead of Greece without any issue?
• Dynamic movement of a circle and resulting ratio of intersecting areas
• A fantasy movie with two races, "Big Ones" (=us) and smaller ones, about saving a newborn baby from a cruel queen
• How can I auto scale fonts imported from mathabx?

more hot questions Question feed Subscribe to RSS Question feed

To subscribe to this RSS feed, copy and paste this URL into your RSS reader.