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Math Help - Help to prove 5^n+5 < 5^(n+1)

  1. #1
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    Help to prove 5^n+5 < 5^(n+1)

    Prove 5^n+5 < 5^(n+1) for al n elements of N

    So i started this by using induction and used n=1 for my base case which i got 10<25 which is true.

    Then i assumed that 5^k+5<5(k+1) for all k elements of N for my induction hypothesis and computed the following as my induction step:

    5(k+1)+5< 5^(k+1)+1
    soo i tried to split the right side to (5^k)x(5^2) then i got stuck
    Can anyone please help me ? Im not sure how to make the right side equal 5^(k+1)
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  2. #2
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    Quote Originally Posted by jess0517 View Post
    Prove 5^n+5 < 5^(n+1) for al n elements of N

    So i started this by using induction and used n=1 for my base case which i got 10<25 which is true.

    Then i assumed that 5^k+5<5(k+1) for all k elements of N for my induction hypothesis and computed the following as my induction step:

    5(k+1)+5< 5^(k+1)+1
    soo i tried to split the right side to (5^k)x(5^2) then i got stuck
    Can anyone please help me ? Im not sure how to make the right side equal 5^(k+1)

    5^{k+1}+5=5\cdot 5^k+5=4\cdot 5^k+\left(5^k+5)<4\cdot 5^k+5^{k+1}<5\cdot 5^k+5^{k+1}=2\cdot 5^{k+1}<5\cdot 5^{k+1}=5^{k+2}

    Tonio
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  3. #3
    A Plied Mathematician
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    Wow. Amazing how many of those inequalities are not tight, and yet you still get the result. The original inequality must be very loose indeed!
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  4. #4
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    Quote Originally Posted by tonio View Post
    5^{k+1}+5=5\cdot 5^k+5=4\cdot 5^k+\left(5^k+5)<4\cdot 5^k+5^{k+1}<5\cdot 5^k+5^{k+1}=2\cdot 5^{k+1}<5\cdot 5^{k+1}=5^{k+2}

    Tonio
    Thank you soo much this was great help.. im just a tad confused how the 4x5^k and the 2x5^k+1 got there.
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  5. #5
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    Thread closed due to this member deleting questions after getting help.
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