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# A couple of bridging-level induction questions watch

1. 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
2. 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)
3. For (a), I'd be inclined to look at , where .

Spoiler:
Show
4. 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
5. Sorry: .
6. 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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Updated: July 20, 2009
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