For n = 2 you can prove it easily.
(a1+a2)/2 > sqrt(a1*a2)
(a1+a2) > 2*sqrt(a1*a2)
(a1+a2)^2 - 4(a1*a2) > 0
(a1-a2)^2 > 0
starting from scratch,
we can apply a few manipulations for convenience...
If the following is true
we attempt to prove that it causes
to be true.
Rewriting for i=1,2,....,n
we try to prove
by showing that it causes
To develop the inequality, the following may be utilised...
This is because
Hence, if the terms are arranged in decreasing order...
Since the terms are arranged in decreasing order, then
If the original statement is true,
Just to note, the above answer does not work. I'm not sure how to exactly solve it, but if you follow the above string of logic, it doesn't work for 2 reasons.
A) You state that "if we assume the sequence decreases" there is nothing in the original problem that says the sequence decreases, only that all numbers of a are greater than 0. they could all be positive and be completely random. It does not make a difference.
b) second of all, if one assumes that a1 is greater than a2 which is greater than a3 etc, then akak+1^k cannot be greater than a1a2....akak+1 simply because akak+1^k has the exact same amount of digits as a1a2.....akak+1, but since a1 is (by transitive property) greater than or equal to ak+1, then the inequality must be false (it can be equal to, but not greater than. If it was simply equal to, it would be a formula, not an inequality, so therefore, that one step does not connect with the previous one)
Other than that, the logic is fine, but since most of the problem only works ASSUMING that the sequence is decreasing, which the problem does not state. If you want to state that the sequence is decreasing, then you would need to prove it somehow. Also, if the sequence is decreasing, then that last chain of logic does not work, as stated above.
On the other hand, I'm not sure how to actually solve it, but I think I'm quite right in noticing the flaw(s) in your logic. I would like to see the actual answer to the problem, please reply!
Oh, by the way, the above Mathematical Induction Problem is hardly Pre-Algebra to Algebra level. I don't even recall doing anything this hard when I did pre-calculus in high school. It should probably be moved to "University Math". Its also the reason why I joined, as I couldn't help but see the flaw, and I felt a need to correct it. I would really like to see a final result though, as I'm not quite sure how to go about solving it either.