Induction Proof that 7^n - 1 is Divisible by 6

We see an easy divisibility proof using induction. Mathematic induction is a tremendously useful proof technique and today we use it to prove that 7^n - 1 is divisible by 6. This is an exercise from Introductory Discrete Mathematics by V.K. Balakrishnan.
Lesson on Induction: • How to Prove by Induct...
A classic induction proof: • Induction Proof for Su...
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[@CovenantOluwaferanmi]

Thank you very much sir. It is even the exact question that is in my handout my lecturer gave us that we couldn't understand what he solved. 😊 Thanks

[@jacksongraham3490]

Thank you so much for the well-explained video!

[@szymon5830]

I mean (a+1)^n-1 = 0 mod(a) proof is trivial using binomial and (a-1)^2n-1= 0 mod(a) if i am not mistaken albo trivial by binomial extension. i found this channel recently and i find it really awesome , greetings from Poland

[@atharvadeshpande1446]

Thank you so much sir for the video. My answer came right.

[@WrathofMath]

I'm glad the video was helpful!

[@mahmoudalbahar1641]

Many thanks for this good video.

[@WrathofMath]

You're welcome and thank you for watching!

[@revitech8378]

Got this one X^(2n -1) + Y^(2n -1) is divisible by (x + y)

[@jboj8430]

I actually solved it when I tried it myself at 1:00 QuQ There is hope for me.

[@WrathofMath]

Awesome work! Now I'll bet you could do this other one easily! kzbin.info/www/bejne/Y52neYiblNOrisk

[@ramaalreem3076]

Which grade induction is for?

[@andresprado1312]

You are a god man, thank you so much

[@vindookiediesel9295]

how do you proof it for all integers n?

[@WrathofMath]

Thanks for watching, I am not sure what you mean. In this video we proved the result for all nonnegative integers. The only other integers are negatives, for which this result does not hold.

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[@samvoor496]

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[@WrathofMath]

I do my best! Thanks for watching!

[@tomkerruish2982]

You could also use the factorization a^n - b^n = (a - b)(a^(n-1) + ... + b^(n-1)) (using a=7, b=1) or the binomial theorem 7^n = (6 + 1)^n = 6^n + n*6^(n-1) + ... + n*6 + 1. Of course, these could (and perhaps should) be established using induction. On a separate note, some of us consider 0 to be a natural number. It might depend on whether we're referencing human nature or Mother Nature.😁