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

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1. Jun 9, 2017

McFluffy

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. Jun 9, 2017

Staff: Mentor

Yes. Only your presentation could be improved. E.g.
$5^{k+1} + 9 < 5^{k+1}+45 = 5\cdot (5^k+9) <_{I.H.} 5 \cdot 6^k < 6 \cdot 6^k = 6^{k+1}$

3. Jun 9, 2017

Buffu

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.

4. Jun 9, 2017

ehild

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

5. Jun 9, 2017

McFluffy

What are the general guidelines that I should follow when writing a proof like this? I'm still quite new to proofs.

6. Jun 9, 2017

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.