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Math Help - Proof by induction

  1. #1
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    Proof by induction

    I need help to solve this question :
    Use induction to prove that, for n>=0:
    3*5^0 + 3*5^1 + 3*5^2 + 3*5^3 + ...+ 3*5^n = 3*(5^(n+1)-1)/4
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by smh745 View Post
    I need help to solve this question :
    Use induction to prove that, for n>=0:
    3*5^0 + 3*5^1 + 3*5^2 + 3*5^3 + ...+ 3*5^n = 3*(5^(n+1)-1)/4
    As 3 is a common factor on both sides it is sufficient to prove:

    5^0 + 5^1 + 5^2 + 5^3 + ...+ 5^n = (5^{n+1}-1)/4

    To start we need to prove that base case (where n=0)

    Then assume it true for n=k, then show that:

    5^0 + 5^1 + 5^2 + 5^3 + ...+ 5^k = (5^{k+1}-1)/4

    implies that:

    5^0 + 5^1 + 5^2 + 5^3 + ...+ 5^{k+1} = (5^{k+2}-1)/4

    You do this by adding 5^{k+1} to both sides of the assumed case, that is:

    (5^0 + 5^1 + 5^2 + 5^3 + ...+ 5^k)+5^{k+1} = (5^{k+1}-1)/4+5^{k+1}

    and show that the right hand side is equal to (5^{k+2}-1)/4

    CB
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  3. #3
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    thanks a lot for your replay

    I appreicated
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