Exploring the Challenges of Mathematical Induction

[mr_coffee]

Hello everyone.

THis is my first proof to strong mathematical induction so im' not sure if its correct or not it seems it though but then again I wrote it. Any suggestions/corrections would be great! THanks

Here it is!
http://suprfile.com/src/1/3j34eh1/lastscan.jpg

 

[Galileo]

Your inductive hypothesis is that $$e_i\leq 3^i$$ for i=1,2,..,k-1, but you used:
$$e_{k-1}\leq 3^k$$
$$e_{k-2}\leq 3^k$$
$$e_{k-3}\leq 3^k$$

That is certainly true, but not strong enough to conclude that
$$e_{k}\leq 3^k$$

What if $$e_{k-1}=e_{k-2}=e_{k-3}= 3^k$$, then your three inequalities hold, but $e_k=3^{k+1}$. You need to use your inductive hypothesis more strongly.

 

[tacman]

Hi there,

First of all, congratulations on tackling strong mathematical induction! It can definitely be a challenging concept to grasp, but with practice and guidance, you will become more comfortable with it.

After reviewing your proof, I have a few suggestions for improvement. Firstly, I would recommend using a more formal language and structure for your proof. This will make it easier for others to follow and understand your reasoning. For example, instead of saying "Let's assume that n is true," you can say "Let n be an arbitrary but fixed integer" or "Let n be a positive integer greater than or equal to 2."

Additionally, I noticed that you used "P(k+1)" in your proof without defining what P(n) is. It's important to clearly define your variables and what they represent in your proof. Also, when proving a statement using strong mathematical induction, you need to show that the statement holds for all integers greater than or equal to your base case. In your proof, you only showed it for n=2 and n=3, but it needs to hold for all integers greater than or equal to 2.

One way to improve your proof is by breaking it down into smaller steps and explaining each step in detail. This will help you and others understand the logic behind your reasoning. Additionally, it's always helpful to include examples or diagrams to illustrate your points.

Overall, keep practicing and seeking feedback from others. Mathematical induction can be challenging, but with persistence and patience, you will master it. Good luck!