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Math Help - problem to prove based on mathematical induction

  1. #1
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    Thumbs up problem to prove based on mathematical induction

    Please give me the answer for the below question.

    prove that (2 pow n . 2 pow n) - 1 is divisible by 3 by mathematical induction.
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  2. #2
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    Quote Originally Posted by sudeepmansh View Post
    Please give me the answer for the below question.

    prove that (2 pow n . 2 pow n) - 1 is divisible by 3 by mathematical induction.
    Where are you having trouble?
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  3. #3
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    Thumbs up

    base step: n=0 : ( 2 pow 0 . 2 pow 0) - 1 = 0 is divisible by 3. so the stmt is true for n=0.

    inductive step: let us assume that the statement is true for any value of n. Now we have to prove that the stmt is true for n = n+1.

    (2 pow n+1 . 2 pow n+1 ) - 1
    2 pow (2n+2) - 1


    how to solve after this step. I am struck here....
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  4. #4
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    Quote Originally Posted by sudeepmansh View Post
    base step: n=0 : ( 2 pow 0 . 2 pow 0) - 1 = 0 is divisible by 3. so the stmt is true for n=0.

    inductive step: let us assume that the statement is true for any value of n. Now we have to prove that the stmt is true for n = n+1.

    (2 pow n+1 . 2 pow n+1 ) - 1
    2 pow (2n+2) - 1


    how to solve after this step. I am struck here....
    Note that 2^{2(k+1)} - 1 = 2^{2k} \cdot 4 - 1 = 4 \left( 2^{2k} - 1\right) + 3.
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