Prove that 7^n - 6n -1 is divisible by 36 for any natural number, n. [Mathematical Induction]

[@jensberling2341]

The love able and lovely performance, and the subtle and clear persuasion of this math. Induction is breathtaking. Yes, breathtaking for me who has struggled to understand it. Thank you, Doctor

[@ayaanmuslimcareafrica6623]

the best way ever to understand Mathematical induction

[@benjaminvatovez8823]

Thank you very much for your video. I would have suggested to use the binomial theorem: 7^n=(6+1)^n=1+6n+36(sum with k from 2 to n) C^k_n 6^(k-2)

[@giuliofalco9816]

Using congruence i proved in this way: 36 = 2^2 3^2 so I apply chinese theorem of remainder writing the following congruence systems: 7^n - 6n - 1 = 0 mod 2; and 7^n - 6n - 1 = 0 mod 3. Substitute 7 with and 6 with the remainder of nteger division by 2 and 3 and the equivalence follow.

[@EzraSchroeder]

this one is straight out of an advanced calculus book haha! it is a problem in chapter one of Kenneth A. Ross: Analysis the Theory of Calculus -- love it!

[@bhaveerathod2373]

YOU ARE GENIUS MAN! THANK YOU, this has helped so much !

[@robertveith6383]

Or, look at (6 + 1)^n - 6n - 1 = 6^n + n*6^(n - 1)*1^1 + [n(n - 1)/2]*6^(n - 2)*1^2 + ... + [n(n - 2)/2]*6^2*1^(n - 2) + n*6^1*1^(n - 1) + 1^n - 6n - 1 = 6^n + n*6^(n - 1) + ... + [n(n - 2)/2]*36 + 6n + 1 - 6n - 1 = 6^n + n*6^(n - 1) + ... + [n(n - 2)/2]*36 That is divisible by 36.

[@gurdiprooprai6453]

Excellent work, but the most difficult part of this video is what determines whether or not you cross your "7's" or not. I am still working on it lol.

[@bhaveerathod2373]

Best teacher

[@KamalAzhar-t7q]

To prove such property :> (where f is a given fonction and p is a fixed integer) one generally can use induction. But the following is much harder

[@lush1523]

Am gonna be here till I finish my degree, I have a discrete mathematics course this semester

[@BP-gn2cl]

Write 7^n =(6+1) ^n, apply bionomial expansion and in fourth line you can prove the required thing

[@ruthgonzalez6547]

Nice!

[@sumeetmahapatra9862]

Thank you sir

[@zazavitch1]

❤

[@aayushixi]

omg tysmmmm

[@abdussamadabubakar1746]

Damn!!!!!