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Math Help - mathematical induction for an inequality

  1. #1
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    mathematical induction for an inequality

    I need to show 1^3 + 2^3 + ... + n^3 < (1/2)n^4 for all n in N and n >=3.

    I want to use mathematical induction, but I don't know if I need to use the first Mathematical induction or the second one?

    Thanks
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  2. #2
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    Quote Originally Posted by inthequestofproofs View Post
    I need to show 1^3 + 2^3 + ... + n^3 < (1/2)n^4 for all n in N and n >=3.

    I want to use mathematical induction, but I don't know if I need to use the first Mathematical induction or the second one?

    Thanks
    If 1^3+2^3+....+n^3 < \frac{1}{2}n^4,\ n\ge3

    then hopefully 1^3+2^3+....+n^3+(n+1)^3 < \frac{1}{2}(n+1)^4

    Proof

    (n+1)^3=n^3+3n^2+3n+1

    (n+1)^4=n^4+4n^3+6n^2+4n+1

    Then

    \frac{1}{2}n^4+(n+1)^3=\frac{1}{2}n^4+n^3+3n^2+3n+  1

    which is less than (n+1)^4

    Hence, if 1^3+2^3+...+n^3 < \frac{1}{2}n^4

    then 1^3+2^3+....+(n+1)^3 is certainly < \frac{1}{2}(n+1)^4

    The inductive process is validated for the hypothesis.

    True for n=3 ?

    1^3+2^3+3^3=36

    \frac{1}{2}3^4=40.5 Proven
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  3. #3
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    We are to prove that 1+2^3+3^3+...+n^3 < \frac{1}{2}n^4, for n\geq 3

    The sum of cube series 1+2^3+3^3+...+n^3 is equal to [\frac{n(n+1)}{2}]^2

    We can restate the question as follows:

    [\frac{n(n+1)}{2}]^2< \frac{1}{2}n^4, for n\geq 3

    Proof:

    Base case: For integer k=3, [\frac{k(k+1)}{2}]^2=36< \frac{1}{2}k^4=\frac{81}{2}


    Induction hypothesis: Suppose for every integer k>3, k\in \mathbb{N}, [\frac{k(k+1)}{2}]^2< \frac{1}{2}k^4

    Then

    (k+1)^3+[\frac{k(k+1)}{2}]^2< \frac{1}{2}(k+1)^4

    LHS: (k+1)^3+[\frac{k(k+1)}{2}]^2

    = k^3+3k^2+3k+1+\frac{k^2(k+1)^2}{4}

    = k^3+3k^2+3k+1+\frac{k^2(k^2+2k+1)}{4}

    = k^3+3k^2+3k+1+\frac{k^4+2k^3+k^2}{4}

    = \frac{k^4}{4}+\frac{3k^3}{2}+\frac{13k^2}{4}+3k+1

    RHS: \frac{1}{2}(k+1)^4

    = \frac{k^4}{2}+2k^3+3k^2+2k+\frac{1}{2}

    Putting LHS and RHS together:

    \frac{k^4}{4}+\frac{3k^3}{2}+\frac{13k^2}{4}+3k+1<  \frac{k^4}{2}+2k^3+3k^2+2k+\frac{1}{2}

    \frac{k^2}{4}+k+\frac{1}{2}<\frac{k^4}{4}+\frac{k^  3}{2}

    Hence, by induction hypothesis, 1+2^3+3^3+...+n^3 < \frac{1}{2}n^4, for all n\geq 3 in \mathbb{N}
    Last edited by novice; February 4th 2010 at 10:28 AM.
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  4. #4
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    question

    After you put the LHS and RHS together, how do you simplify the inequality?\

    thanks for your detailed work
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  5. #5
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    Simple algebra. Move the 1st and 2nd terms to the right hand side, and move the last term from the right hand side to the left, etc.
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  6. #6
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    question about inequality

    THanks!! it makes sense. My new question is that if what we are trying to prove the < (inequality), it seems that the RHS < LHS, because some terms are larger than others. However, there are other terms that are smaller than others. So, even though it seems logical that the inequality is true, is it enough to state it like that, or is there a need of more explanation to justify the inequality?
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  7. #7
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    Quote Originally Posted by inthequestofproofs View Post
    THanks!! it makes sense. My new question is that if what we are trying to prove the < (inequality), it seems that the RHS < LHS, because some terms are larger than others. However, there are other terms that are smaller than others. So, even though it seems logical that the inequality is true, is it enough to state it like that, or is there a need of more explanation to justify the inequality?
    That's all.

    If the base case and induction hypothesis are true, then proposition is true for all positive integers---end of proof.
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  8. #8
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    Quote Originally Posted by novice View Post
    Putting LHS and RHS together:

    \frac{k^4}{4}+\frac{3k^3}{2}+\frac{13k^2}{4}+3k+1<  \frac{k^4}{2}+2k^3+3k^2+2k+\frac{1}{2}

    \frac{k^2}{4}+k+\frac{1}{2}<\frac{k^4}{4}+\frac{k^  3}{2}

    Hence, by induction hypothesis, 1+2^3+3^3+...+n^3 < \frac{1}{2}n^4, for all n\geq 3 in \mathbb{N}
    If all the steps to here are understandable, inthequestforproofs,

    then you can say \frac{k^4}{4} > \frac{k^2}{4}

    definately k^4 > k^2 for k >1

    Also, we can ask... is \frac{k^3}{2} > k+0.5 ?

    So, is \frac{k^3}{2} > \frac{2k+1}{2} ?

    Is k^3 > 2k+1 ?

    Is k(k^2) > 2(k+0.5) ?

    if k > 2, then k^2 > k+0.5
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