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Math Help - Looking for patterns

  1. #1
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    Looking for patterns

    Hey guys, I have a question where the book asks me to produce a pattern after observing the difference in the changing numbers. This completely stumped me. Here's the question:

    The numbers 1, 8, 27, and 64 are the first 4 cubes.

    a) Find the sum of the first 2 cubes. ans. 9
    b) Find the sum of the first 3 cubes. ans. 36
    c) Find the sum of the first 4 cubes. ans. 100
    d) Describe the pattern in the sums.
    e) Use the pattern to find the sum of the first 9 cubes.

    So I determined the sums but I couldn't develop an expression that explains the pattern. I already looked at the resulting answer, but I'm not even sure what the hell I'm supposed to do to even get close to the answer. Am I supposed to use trial and error?
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  2. #2
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    Re: Looking for patterns

    Quote Originally Posted by darksoulzero View Post
    Hey guys, I have a question where the book asks me to produce a pattern after observing the difference in the changing numbers. This completely stumped me. Here's the question:

    The numbers 1, 8, 27, and 64 are the first 4 cubes.

    a) Find the sum of the first 2 cubes. 9
    b) Find the sum of the first 3 cubes. 36
    c) Find the sum of the first 4 cudes. 100
    d) Describe the pattern in the sums.
    e) Use the pattern to find the sum of the first 9 cubes.

    So I determined the sums but I couldn't develop an expression that explains the pattern. I already looked at the resulting answer, but I'm not even sure what the hell I'm supposed to do to even get close to the answer. Am I supposed to use trial and error?
    3^2 , 6^2 , 10^2 , ...

    would you agree the next term is 15^2 ?
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    Re: Looking for patterns

    Quote Originally Posted by skeeter View Post
    3^2 , 6^2 , 10^2 , ...

    would you agree the next term is 15^2 ?
    If the pattern is what I think it is, then yeah, it's 15^2. Is the next term 21^2?
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    Re: Looking for patterns

    Quote Originally Posted by darksoulzero View Post
    If the pattern is what I think it is, then yeah, it's 15^2. Is the next term 21^2?
    now that you've determined a pattern, what is the sum ...

    1^3 + 2^3 + 3^3 + 4^3 + ... + n^3 = \sum_{k=1}^n k^3

    ?
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    Re: Looking for patterns

    Quote Originally Posted by darksoulzero View Post
    If the pattern is what I think it is, then yeah, it's 15^2. Is the next term 21^2?
    Can you use induction to prove this?:

    \sum\limits_{k = 1}^N {k^3 }  = \left( {\frac{{N^2  + N}}{2}} \right)^2,~~N\ge 1
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  6. #6
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    Re: Looking for patterns

    Quote Originally Posted by skeeter View Post
    now that you've determined a pattern, what is the sum ...

    1^3 + 2^3 + 3^3 + 4^3 + ... + n^3 = \sum_{k=1}^n k^3

    ?
    Ohh, I see now. Hmm, but the pattern at the back of my book is described as
    \frac{n^2(n + 1)^2}{4}

    And I don't understand what that means.

    Quote Originally Posted by Plato View Post
    Can you use induction to prove this?:

    \sum\limits_{k = 1}^N {k^3 }  = \left( {\frac{{N^2  + N}}{2}} \right)^2,~~N\ge 1
    Hmm, I don't think so. I tried to,but then I realized that I don't know how to prove something. Can I prove it's true by showing that the numbers 1-4 correlate to the numbers of the first 4 cubes? Or does it correlate to the sum of the numbers before N including the Nth term?
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  7. #7
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    Re: Looking for patterns

    here is the sequence of partial sums (w/o the squares, ignore that for the meantime)

    1, 3, 6, 10, 15, 21, ...

    double this sequence ...

    2, 6, 12, 20, 30, 42, ...

    note how each term of the sequence factors (see the pattern?)

    (1 \cdot 2) , (2 \cdot 3) , (3 \cdot 4), (4 \cdot 5) , (5 \cdot 6) , (6 \cdot 7) , ...

    "un" double ...

    \frac{1}{2} \left[(1 \cdot 2) , (2 \cdot 3) , (3 \cdot 4) , (4 \cdot 5) , (5 \cdot 6) , (6 \cdot 7) , ...\right]
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