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anonymous

  • one year ago

Show that (n^2) + 4 < (n + 1)^2 for all natural numbers n>=2

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  1. anonymous
    • one year ago
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    for \(n \geq 2\) 4 < 2n +1, right?

  2. anonymous
    • one year ago
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    I plugged in 2 for n to show the base case.

  3. anonymous
    • one year ago
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    base case n =2, 4 < 2n +1 =5, check the left hand side is 4, a constant, never change. while the right hand side is increasing with n > 2

  4. anonymous
    • one year ago
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    For example, if n =3 , 2n +1 = 7 >4 n =4, 2n +1 = 9 >4 and so on. We all have 2n +1 >4

  5. anonymous
    • one year ago
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    This should be a mathematical induction proof.

  6. anonymous
    • one year ago
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    no need

  7. anonymous
    • one year ago
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    I mean it is required for my homework.

  8. anonymous
    • one year ago
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    oh!! no need to use induction for this because it is so simple to get the proof

  9. anonymous
    • one year ago
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    ok, I show you this first 4 < 2n +1 +n^2 both sides you have n^2 +2 < n^2 +2n +1 the right hand side is (n+1)^2 done.

  10. anonymous
    • one year ago
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    Now, if you want to use induction, ok, go ahead basic case n =2 n^2 +4 = 8 < (n+1)^2 =(2+1)^2 =9 check

  11. anonymous
    • one year ago
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    Suppose it holds for n = k , that is k^2 + 4 < (k+1)^2

  12. anonymous
    • one year ago
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    Now prove it holds for n = k+1 that is (k+1)^2 +4 < ((k+1)+1)^2

  13. anonymous
    • one year ago
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    Yeah, this is where I got stuck.

  14. jim_thompson5910
    • one year ago
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    It might help to think of `k^2 + 4 < (k + 1)^2` as `k^2 + 4 + q = (k + 1)^2` where `q` is some positive number

  15. anonymous
    • one year ago
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    ok, from k^2 +4 < k^2 +2k+1 add 2k +1 both sides k^2 + 2k +1 +4 < k^2 + 2k +1 +2k +1 (k+1)^2 +4 < k^2 +4k +2 If we consider (k+2)^2 = k^2 + 4k + 4 which is greater than our left hand side

  16. anonymous
    • one year ago
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    hence \((k+1)^2 +4< k^2 +4k +2 < k^2 +4k +4\) pick far left and far right, we have what we need to prove. ok?

  17. anonymous
    • one year ago
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    sorry for above, greater than our RIGHT hand side

  18. anonymous
    • one year ago
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    |dw:1442708748614:dw|

  19. anonymous
    • one year ago
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    Thanks

  20. anonymous
    • one year ago
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    I understood it.

  21. anonymous
    • one year ago
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    But, how did u know to add 2k + 1 to both sides?

  22. anonymous
    • one year ago
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    trick: open the left hand side of the expression we need to prove, that is (k+1) ^2 +4 = k^2 +2k +1 +4 we can see that we need 2k +1

  23. jim_thompson5910
    • one year ago
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    Here's an alternative method to do the inductive step. I'm attaching it as a txt file so I don't barge in too much and add more clutter to the page.

  24. anonymous
    • one year ago
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    Thanks, to both of you.

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