Induction Proof that 7^n - 1 is Divisible by 6
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- เผยแพร่เมื่อ 14 ต.ค. 2024
- 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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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
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
Thank you so much for the well-explained video!
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.😁
Got this one X^(2n -1) + Y^(2n -1) is divisible by (x + y)
Thank you so much sir for the video. My answer came right.
I'm glad the video was helpful!
I actually solved it when I tried it myself at 1:00 QuQ
There is hope for me.
Awesome work! Now I'll bet you could do this other one easily! th-cam.com/video/-lDHUe_nxTg/w-d-xo.html
Which grade induction is for?
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Many thanks for this good video.
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how do you proof it for all integers n?
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.