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Prove ##5^n+9<6^n## for ##n\epsilon N|n\ge2## by induction

  1. Jun 9, 2017 #1
    1. The problem statement, all variables and given/known data
    Prove ##5^n+9<6^n## for ##n\epsilon \mathbb{N}|n\ge2## by induction.

    2. Relevant equations
    None

    3. The attempt at a solution
    The base case which is when ##n=2##:

    ##5^2+9<6^2##
    ##34<36##

    Thus, the base case is true. Now for the induction step.

    Induction hypothesis: Assume ##5^k+9<6^k## for ##k \epsilon \mathbb{N}|k\ge2##

    Induction step: To show that ##5^k+9<6^k## implies ##5^{k+1}+9<6^{k+1}##, I'll start with the induction hypothesis:

    ##5^k+9<6^k##
    ##5(5^k+9)<6(6^k)## since we have an inequality and not an equation, I can add numbers that are bigger to one side and not violate inequality. With this in mind, since ##5<6##, I'd multiplied 5 on the smaller side and 6 on the bigger side.

    ##5^{k+1}+45<6^{k+1}##
    ##5^{k+1}+9<6^{k+1}##

    Multiply one side by a bigger number than the other side is what makes this proof possible. And I think this is a logical thing to do because we're dealing with an inequality and not an equation. Are my steps justified?
     
  2. jcsd
  3. Jun 9, 2017 #2

    fresh_42

    Staff: Mentor

    Yes. Only your presentation could be improved. E.g.
    ##5^{k+1} + 9 < 5^{k+1}+45 = 5\cdot (5^k+9) <_{I.H.} 5 \cdot 6^k < 6 \cdot 6^k = 6^{k+1}##
     
  4. Jun 9, 2017 #3
    Yes it is all correct. Multiplication here is just adding and you can always add smaller term to a side and larger term to other side without flipping the sign.

    Also you may want to use ##\in## (\in) instead of ##\epsilon##.

    Sorry I did not see the post by @fresh_42 , it did not get updated on page.
     
  5. Jun 9, 2017 #4

    ehild

    User Avatar
    Homework Helper
    Gold Member

    Yes, but you can show it mathematically, with less words: ##5(5^k+9)<5(6^k)<6^{k+1}##

    Also, you can write in the last step ##5^{k+1}+9<5^{k+1}+45<6^{k+1}##
     
  6. Jun 9, 2017 #5
    What are the general guidelines that I should follow when writing a proof like this? I'm still quite new to proofs.
     
  7. Jun 9, 2017 #6

    fresh_42

    Staff: Mentor

    Good question. I think the important part is to always make clear what follows from what. Start with what you have, in this case ##5^k+9<6^k## and either ##5^{k+1}+9## and make it bigger, or ##6^{k+1}## and make it smaller. Personally I find it easier to read when it get's bigger, but this is a matter of taste. If you write your expressions from one line to the next and always handle both sides, it is more difficult to see whether you follow a path or simply write down some expressions. If both sides ##5^{k+1}+9## and ##6^{k+1}## appear in the same line, then it's hard to tell, whether you still deal with the claim or already entered the proof.
    Of course you can do this in a draft and order the terms afterwards for the final version. Imagine if it were you to read it. A proof should lead from what is given to what has to be shown and the steps in between should show the implications needed. As I said, it wasn't wrong what you've written, only a bit confusing to read. You have @ehild's and mine example. Compare them with yours.

    As I learnt it, I followed a short template.

    Condition: Given facts.
    Statement: What has to be shown.
    Proof: .... and in case it was an indirect proof, I wrote. Assumption: (the opposite of what has to be shown)
    Now come the steps ##\Rightarrow## ... ##\Rightarrow## ...
    And at the end a sign that it is the end, like a small box or so. And sure, I abbreviated these underlined words.
     
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