What are the common challenges faced in strong 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.