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Math Help - Induction proof

  1. #1
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    Induction proof

    hey all hope ur fine,
    i have a homeworks really hard bcz i was absent for 2 weeks(i was sick)
    so plz can u help me??

    the lesson name: Recurrence

    1) n being an entire naturel,proove that:
    3^(2n)_ 2^(n) divisible by 7
    2)5^(2n)_2^(n) divisible by 23

    -------------------------
    proove one polynome p(x)=ax^2+bx that verifie,for all reel x's, p(x+1)_p(x)=x

    etablissez by recurrence that p(n) entire east for all n E N

    be the term continuation generale Sn=1+2+......+n, n_>1

    hey sorry for my english iam learning math in french but this forum da best many thanks
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  2. #2
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    Hello, iceman1!

    I can help with the first problem . . .


    1) n is a natural number. .Prove that:

    (a) 3^{2n} - 2^n is divisible by 7

    (b) 5^{2n} - 2^n is divisible by 23

    (a) We have: .N .= .(3)^n - 2^n .= .9^n - 2^n .= .(7 + 2)^n - 2^n

    Expand the binomial:
    N .= .
    [7^n + n7^{n-1}2 + [n(n-1)/2]7^{n-2}2 + . . . + n72^{n-1} + 2^n] - 2^n

    And we have:
    N .= .7^n + n7^{n-1}2 + [n(n-1)/2]7^{n-2}2 + ... + n72^{n-1}

    Since every term has a factor of 7, N is divisible by 7.



    (b) We have: .N .= .(5)^n - 2^n .= .25^n - 2^n .= .(23 + 2)^n - 2^n

    Now proceed as in part (a).

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