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Thread: Mathematical Induction

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    Cadet JustRandomThings's Avatar
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    Mathematical Induction

    Hey every 1 ty in advance ^^

    Prove that 3^(3n) + 2^(n+2) is divisible by 5 for all integers n greater than or equal to 1
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    I love trials pikachu975's Avatar
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    Re: Mathematical Induction

    3^(3n) + 2^(n+2)

    Step 1: Prove true for n=1
    LHS = 3^(3) + 2^(1+2)
    = 27 + 8
    = 35 which is divisible by 5, therefore true for n=1

    Step 2: Assume true for n=k
    3^(3k) + 2^(k+2) = 5M (where M is an integer)

    Step 3: Prove true for n=k+1
    LHS = 3^(3(k+1)) + 2^(k+3)
    = 3^(3k+3) + 2^(k+2+1)
    = 3^3 * 3^(3k) + 2 * 2^(k+2)
    = 27 (3^(3k) + 2^(k+2)) - 25(2^(k+2)) ----- I factorised 27(2^(k+2)) so I had to minus 25 of it to pay it back
    = 27 (5M) - 25(2^(k+2)) ----- using assumption
    = 5 (27M - 5(2^(k+2))) and since 27, M, 5, and 2^(k+2) are integers,
    then it is true for n=k+1

    Step 4: Conclusion
    By mathematical induction, 3^(3n) + 2^(n+2) is divisible by 5 for n >= 1


    Soz if it's hard to read it'd take ages to type this in latex
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    Ancient Orator leehuan's Avatar
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    Re: Mathematical Induction

    Quote Originally Posted by pikachu975 View Post
    Soz if it's hard to read it'd take ages to type this in latex
    Not really

    Just replace all the parentheses ( ) with braces { }

    And use \\ to change lines instead
    Last edited by leehuan; 7 Feb 2017 at 12:28 PM.

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    Re: Mathematical Induction

    Quote Originally Posted by leehuan View Post
    Not really

    Just replace all the brackets ( ) with braces { }

    And use \\ to change lines instead
    ( ) are parentheses; brackets are [ ].

    Of course we often refer to ( ). [ ] and { } simply as brackets.
    Last edited by Drongoski; 7 Feb 2017 at 9:22 AM.
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    Ancient Orator leehuan's Avatar
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    Re: Mathematical Induction

    Quote Originally Posted by Drongoski View Post
    ( ) are parentheses; brackets are [ ].

    Of course we often refer to ( ). [ ] and { } simply as brackets.
    Fixed. I forgot the word at the time.

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