#20 prove induction n^3- n is divisible by 3 mathgotserved mathematical precalculus discrete princ

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Clear, slow and explains alternative methods. Thanks so much <3.

brandenvs

I came up with a very simple solution.
First, n^3 - n = n(n^2-1) = n(n+1)(n-1) = (n-1)n(n+1)
We have 3 cases, either n is writable by the form 3k, 3k+1 or 3k+2.
If n is writable by the form 3k, it satisfies our proposition.
If n is writable by the form 3k+1, then n-1 is writable by the form 3k, therefore, n^3 - n is divisible by 3.
Otherwise, n is writable by the form 3k+2, then n+1 is writable by the form 3k, '...'.
CQD.

Lucas-hrmj

n^3 - n = n(n^2 - 1) = (n - 1)n(n + 1). We have 3 consecutive natural numbers. Therefore exactly one of them is divisible by 3.

Therefore n^3 - n is divisible by 3.

sanelprtenjaca

Really helpful to me.Thank you so much🙏🙏🙏

snehaar

nice video Please affirmation this question, show that n^2+2n is divisible by 3 for all n≥1 (by using mathematical induction )

MikiasZerihun-zhpw

tysm, my teacher said to complete this proof as homework but he did only 2 induction proofs with us in class, so im just getting used to proofs, but you helped a lot.

near

Nice, but the video was slow made me doze off 😅. I fixed that by increasing the speed to 1.5×. I don't think you should increase the speed thou, it takes time to grasp the concepts.

edwinmifatu

You saved my life, May God Bless Thee 😘

juraijbinjamal

Great video.... but n^3-n is also divisible by six. There's an easy logic reasoning in that n^3-n can be factorised into n(n-1)(n+1) which describes 3 consecutive numbers and considering 3 consecutive numbers must have an even number and a multiple of 3, the product of 3 consecutive numbers must be divisible by 6. How would you prove this with your method though?

jacobkelly

you can also do it this way: n³-n=n(n-1)(n+1). Then split it into cases n=3k, n=3k+1, n=3k-1 to see that all three will give 3 times an integer

helloitsme

Hi, I have a question about the proof by induction here. Here you stated that n is greater than or equal to 1. Cant n also be less than or equal to one as I am thinking about the negative numbers that are divisible by 3?...Thank you

peterkrahn

Prove by induction that n3−n is divisible by 24 when pls solve

sammuavkinlife

Let's say n is dividable by 3
Therefore n^3-n is dividable by 3
(n+1)^3- n -1=n3+3n2+3n+1-n-1
That's dividable by 3
N-1 is kinda the same

xxpod

For n=1, the value of k is zero. However, you said k is a positive integer. How would you explain it?

shanmugarajah

yo, 1^3 isnt 3 though in the base case... Or am I missing something?

iScreamRules

So If like this n^4-4n^2 by 3 ? How to?

demonsj

11:03 how to come this equation. i could not understand it?

dailyeyesonfakharzaman

there is a much simpler way to prove this using geometry

gregor

#20 prove induction n^3- n is divisible by 3 mathgotserved mathematical precalculus discrete princ

#20 prove induction n^3- n is divisible by 3 mathgotserved mathematical precalculus discrete princ

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