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Math Help - Finding if this recursively defined sequence satisfies this explicit formula

  1. #1
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    Finding if this recursively defined sequence satisfies this explicit formula

    Hey Guys.

    Having a bit of trouble with recursion and seeing if the sequence satisfies the formula.

    I need to determine whether

    ak=2ak-1+k-1 for all integers k>=2

    satisfies the explicit formula an=(n-1)2 for all integers n>=1

    Not quite sure on what the right steps are.

    Thanks for your help in advance!
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  2. #2
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    Re: Finding if this recursively defined sequence satisfies this explicit formula

    Hey RadMabbit.

    What does the 2_(ak-1) mean? Is this 2 * a_(k-1)?
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  3. #3
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    Re: Finding if this recursively defined sequence satisfies this explicit formula

    Sorry, typo

    it is 2*ak-1

    so 2 multiplied by the previous a term.
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  4. #4
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    Re: Finding if this recursively defined sequence satisfies this explicit formula

    This is just a substitution where if a_k = (k-1)^2 then a_(k-1) = ((k-1) - 1)^2 = (k-2)^2.

    Now its just a matter of checking whether LHS = RHS with that substitution.
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  5. #5
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    Re: Finding if this recursively defined sequence satisfies this explicit formula

    would you care to elaborate.
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  6. #6
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    Re: Finding if this recursively defined sequence satisfies this explicit formula

    So a_k = (k-1)^2, a_(k-1) = (k-2)^2

    So 2*a_(k-1) + (k-1)
    = 2*(k-2)^2 + k - 1
    = 2k^2 - 8k + 8 + k - 1
    = 2k^2 - 7k + 7
    != (k-1)^2 = k^2 - 2k + 1 in general.

    To check when they are equal equate the two and you get the condition:

    k^2 - 5k + 6 = 0 or (k-2)(k-3) = 0 so if k = 2 or k = 3 then the equality holds but other-wise it doesn't.
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  7. #7
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    Lightbulb Re: Finding if this recursively defined sequence satisfies this explicit formula

    a_k=1-2 k+k^2
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