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anonymous

  • one year ago

mathematical induction to prove that n 3 − n is divisible by 3, for every positive integer n

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  1. SolomonZelman
    • one year ago
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    \(n^3-n=n(n^2-1)=n(n-1)(n+1)\) that should help.

  2. anonymous
    • one year ago
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    thank you

  3. SolomonZelman
    • one year ago
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    No further questions?

  4. anonymous
    • one year ago
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    not yet... im working it thru

  5. anonymous
    • one year ago
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    ok this is where im at P(x+1)=(x+1)((x+1)−1)((x+1)+1) = 3m do i divide

  6. SolomonZelman
    • one year ago
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    you don't need anything, except for a little logic.

  7. SolomonZelman
    • one year ago
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    here, tell me what happens if you have a product that consists of integers and one of these integers are divisible by 3? Do you agree that the result is divisible by 3?

  8. anonymous
    • one year ago
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    yes

  9. SolomonZelman
    • one year ago
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    Ok, good.

  10. SolomonZelman
    • one year ago
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    Now, lets come back to the fact that you want to prove that: *n(n-1)(n+1)* is divislbe by 3, \(\forall {\bf n \in \mathbb Z}\)

  11. SolomonZelman
    • one year ago
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    Ok, lets consider 3 possible cases (for possible integer k) 3k, 3k+1, and 3k+2

  12. SolomonZelman
    • one year ago
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    If your number n falls under the category 3k (Such that n --> 3k) then the *n* component of *n(n-1)(n+1)*, is divisible by 3, and thus the entire product *n(n-1)(n+1)* is divislbe by 3.

  13. SolomonZelman
    • one year ago
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    If your number n falls under the category 3k+1 (Such that n --> 3k+1) then the *n-1* component of *n(n-1)(n+1)*, is divisible by 3, and thus the entire product *n(n-1)(n+1)* is divislbe by 3.

  14. SolomonZelman
    • one year ago
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    And then if: n --> 3k+2 then the *n+1* makes it divisible by 3.

  15. SolomonZelman
    • one year ago
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    Do I sound rediculous, or is it understandable.

  16. anonymous
    • one year ago
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    no u dont.. im just trying to absorb the concept

  17. anonymous
    • one year ago
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    my brain is a bit mathed out

  18. SolomonZelman
    • one year ago
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    \(\displaystyle\int\)\(\theta\) \(f(u)+nn=y\)

  19. SolomonZelman
    • one year ago
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    If you have more questions to ask, I wil be back if I am online.

  20. anonymous
    • one year ago
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    thank you

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