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Math Help - difference of two nth powers (proof)

  1. #1
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    difference of two nth powers (proof)

    a^1 - b^1 = (a - b)
    a^2 - b^2 = (a-b)(a+b)
    a^3 - b^3 = (a-b)(a^2 + ab + b^2)
    a^4 - b^4 = (a-b)(a^3 + a^2b + ab^2 + b^3)

    If you continue with increasing the exponent then you notice the following general rule: a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + ... + ab^{n-2} + b^{n-1})
    However, this doesn't proof that this rule is valid for every positive integer exponent. How can you proof that this rule indeed is valid for every positive integer exponent?
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  2. #2
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    Quote Originally Posted by Bart View Post
    However, this doesn't proof that this rule is valid for every positive integer exponent. How can you proof that this rule indeed is valid for every positive integer exponent?
    Induction?
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  3. #3
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    No need for induction. Just define \displaystyle n to be a positive integer.

    Then \displaystyle (a - b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \dots + a^2b^{n-3} + a\,b^{n-2} + b^{n-1})

    \displaystyle = a(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \dots + a^2b^{n-3} + a\,b^{n-2} + b^{n-1}) - b(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \dots + a^2b^{n-3} + a\,b^{n-2} + b^{n-1})

    \displaystyle = (a^n + a^{n - 1}b + a^{n-2}b^2 + \dots + a^3b^{n - 3} + a^2b^{n - 2} + a\,b^{n - 1}) - (a^{n-1}b + a^{n-2}b^2 + a^{n-3}b^3 + \dots + a^2b^{n-2} + a\,b^{n-1} + b^{n})

    \displaystyle = a^n - b^n (since all the middle terms cancel).


    Q.E.D.
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    Quote Originally Posted by Prove It View Post
    No need for induction. Just define \displaystyle n to be a positive integer.
    \displaystyle = a^n - b^n (since all the middle terms cancel).
    May I ask you this: without induction how do we know that those “middle terms cancel”
    I submit that you do not know that.
    Actually this a beautifully simple induction question.
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  5. #5
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    Quote Originally Posted by Plato View Post
    May I ask you this: without induction how do we know that those “middle terms cancel”
    I submit that you do not know that.
    Actually this a beautifully simple induction question.
    I submit that it is known because it is obvious...
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  6. #6
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    Plato, I see your point but I also see Prove It's. You can compare the "middle terms" term by term and see that they cancel in pairs. Of course, you have to assume that the terms covered by the "..." will do the same.
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  7. #7
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    Question

    Quote Originally Posted by Bart View Post
    a^1 - b^1 = (a - b)
    a^2 - b^2 = (a-b)(a+b)
    a^3 - b^3 = (a-b)(a^2 + ab + b^2)
    a^4 - b^4 = (a-b)(a^3 + a^2b + ab^2 + b^3)

    If you continue with increasing the exponent then you notice the following general rule: a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + ... + ab^{n-2} + b^{n-1})
    However, this doesn't proof that this rule is valid for every positive integer exponent. How can you proof that this rule indeed is valid for every positive integer exponent?
    Can this be proven using the binomial theorem?
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  8. #8
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    Quote Originally Posted by wonderboy1953 View Post
    Can this be proven using the binomial theorem?
    Sure. Let c = a - b

    a^n - b^n = (b + c)^n - b^n = \sum_{k=0}^n \binom{n}{k}b^kc^{n-k} - b^n<br />
= \sum_{k=0}^{n-1} \binom{n}{k}b^kc^{n-k}<br />
    <br />
= c\sum_{k=0}^{n-1} \binom{n}{k}b^kc^{n-k-1}
    Now substitue back in for c.
    = (a-b)\sum_{k=0}^{n-1} \binom{n}{k}b^k(a - b)^{n-k-1}
    = (a-b)\sum_{k=0}^{n-1} \binom{n}{k}b^k<br />
\sum_{l=0}^{n-k-1}\binom{n-k-1}{l}a^lb^{n-k-l-1}
    = (a-b)\sum_{k=0}^{n-1}\sum_{l=0}^{n-k-1}\binom{n}{k}\binom{n-k-1}{l}a^lb^{n-l-1}
    ...
    You can keep going, but it seems like the other way is easier
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  10. #10
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    Quote Originally Posted by TheCoffeeMachine View Post
    Nice rule
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