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  1. #1
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    Mathematical Induction

    Prove by mathematical induction that 7^(2n) - 48n - 1 is a multiple of 2304

    I am ok with the first and second steps, but I am confused with the third i.e. when substituting n = k + 1.

    Any help please?
    Thanks a lot
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by yobacul View Post
    Prove by mathematical induction that 7^(2n) - 48n - 1 is a multiple of 2304

    I am ok with the first and second steps, but I am confused with the third i.e. when substituting n = k + 1.

    Any help please?
    Thanks a lot
    Hint:

    Note that 7^{2(k + 1)} - 48(k + 1) - 1 = 7^{2k + 2} - 48k - 48 - 1 = 7^{2k + 2} ~\underbrace{{\color{red} - 7^2 \cdot 48k + 7^2 \cdot 48k}}_{\text{this is zero}}~ - 48k - 7^2
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  3. #3
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    Quote Originally Posted by yobacul View Post
    Prove by mathematical induction that 7^(2n) - 48n - 1 is a multiple of 2304

    I am ok with the first and second steps, but I am confused with the third i.e. when substituting n = k + 1.

    Any help please?
    Thanks a lot
    hi yobacul,

    F(n)

    7^{2n}-48n-1

    is a multiple 2304 ?

    F(k+1)

    7^{2(k+1)}-48(k+1)-1

    is a multiple of 2304 if the "k"th term is ?

    Does the hypothesis F(n) cause this to be true ?

    7^{2k+2}-48k-48-1=7^2\left(7^{2k}\right)-48k-49

    =7^2\left(7^{2k}\right)-48k-7^2=7^2\left(7^{2k}-1\right)-48k

    Now express -48k as a multiple of 7^2

    To do this we need to subtract another 48(48k)
    {and therefore also add that amount}

    7^2\left(7^{2k}-1\right)-(48k)-48(48k)+48(48k)

    =7^2\left(7^{2k}-1\right)-49(48k)+48(48k)

    =7^2\left(7^{2k}-48k-1\right)+48^2k

    The final term is 48^2k=2304k

    therefore, the term-by-term link is established.

    Hence, if F(n) is valid, F(n+1) also is

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