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Math Help - Help finishing a proof

  1. #1
    Junior Member
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    Oct 2009
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    Help finishing a proof

    Could someone help me finish this proof? Everything makes sense to me up to the last part... I'm not sure how to prove that 5d_{k} - 6d_{k-1}=2^{k+1} + 3^{k+1}
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  2. #2
    Super Member
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    Israel
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    Hi!

    If we look at the LHS, it is: 2^{k+1}+3^{k+1} = 2\cdot 2^k + 3 \cdot 3^k

    RHS: 5d_k -6d_{k-1} = 5(2^k+3^k) -6(2^{k-1}+3^{k-1}) = 5\cdot 2^k +5\cdot 3^k -6\cdot 2^{k-1}-6\cdot 3^{k-1} = 5\cdot 2^k +5\cdot 3^k - 3\cdot 2^k -2 \cdot 3^k = 2\cdot 2^k + 3\cdot 3^k

    And that is equal to the LHS, and so we are done.
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  3. #3
    Junior Member
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    Oct 2009
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    I spent three hours banging my head on that problem tonight, I really appreciate your response. I understand how it works now. Thanks so much!
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