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    I am currently studying the Martin Liebeck book; A concise introduction to pure mathematics, but unfortunately, as some people may be aware, he does not include solutions to the problems at the end of each chapter. There are a couple of induction questions which I have been hacking away at for a couple of days now, but think I might need a push in the right direction! Here goes;

    Prove:

    (a) For all integers n>=0, the number (5^2n)-(3^n) is a multiple of 11.

    (b) If x>=2 is a real number and n>=1 is an integer, then (x^n)>=nx.

    (c) If n>=3 is an integer, then (5^n) > (4^n)+(3^n)+(2^n).

    I am completely familiar with the principle of induction, but the inductive step in each case (showing that P(n) => P(n+1)), I have found difficult.
    (b) is of particular importance as I have read that it is an important result.

    Any help much appreciated, thanks,

    Nathan
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    Hints:

    (a) 5^2n = 25^n (which also = (22+3)^n).
    (b) if x>=2, n>=1, then xn >= x(n+1)
    (c) multiply both sides by 5, then show the RHS is > 4^(n+1) + 3^(n+1) + 2^(n+1)
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    For (a), I'd be inclined to look at u_{n+1} - u_n, where u_n = 5^{2n} - 3^n = 11k.

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    \newline u_{n+1} - u_n = 5^{2n+2} - 5^{2n} + 3^n - 3^{n+1}\newline

= 24.5^{2n} - 2.3^n\newline

= 22.5^{2n} - 2.3^n + 2.5^{2n}\newline

= 22.5^{2n} + 2u_n\newline

= 2.11.5^{2n} + 2.11k\newline

\Rightarrow u_{n+1} = 2.11.5^{2n} + 2.11k + u_n = 11\big[ 2.5^{2n} + 3k \big]
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    I have had a quick go at (a), and I understand that now, although I never would have come up with nuodai's method.
    Forgive me if I'm wrong, but for (b), DFranklin, you have stated that xn >= x(n+1) which is incorrect (as far as I can see). This does yield the correct answer but I don't see that is correct.
    The last one I shoul be able to do, I will try again after dinner!

    Thanks very much guys!

    Nathan
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    Sorry:  xn \ge (n+1).
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    Thanks that's better! I have done all three now they aren't that hard in hindsight. Think I may just need to spend more time looking at replacing expressions within the inequalities to transform them correctly.

    Thank you very much for your help!

    Nathan
 
 
 

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