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Thread: 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?
    $\displaystyle 3^2 , 6^2 , 10^2 , ...$

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

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

    would you agree the next term is $\displaystyle 15^2$ ?
    If the pattern is what I think it is, then yeah, it's $\displaystyle 15^2$. Is the next term $\displaystyle 21^2$?
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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 $\displaystyle 15^2$. Is the next term $\displaystyle 21^2$?
    now that you've determined a pattern, what is the sum ...

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

    ?
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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 $\displaystyle 15^2$. Is the next term $\displaystyle 21^2$?
    Can you use induction to prove this?:

    $\displaystyle \sum\limits_{k = 1}^N {k^3 } = \left( {\frac{{N^2 + N}}{2}} \right)^2,~~N\ge 1 $
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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 ...

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

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

    And I don't understand what that means.

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

    $\displaystyle \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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    Re: Looking for patterns

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

    $\displaystyle 1, 3, 6, 10, 15, 21, ...$

    double this sequence ...

    $\displaystyle 2, 6, 12, 20, 30, 42, ...$

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

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

    "un" double ...

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