Re: HSC 2015 4U Marathon
Here's my solution for part (i) and an idea for part (ii) - it's probably not written in the most mathematically concise way but it's something:
https://imgur.com/kldJ69o
I may be wrong but I feel like there is some sort of circularity here:
It might be a misinterpretation but it seems you assume that x(0) is the lowest of the x(n) which from there allows you to prove that x(n+1) > x(n).
So it is either circular in that you assume x(n+1) > x(n) to get that min(x(n)) = x(0),
or, you have independent reasons for knowing that min(x(n)) = x(0), in which case what would you say to prove this?
What if because of the 1/5 in front of the bracket that there is some value below x(0)?
The general gist of it is true however (the reasoning prior to this), and to complete the proof safely, I would recommend simply using proof by induction
Step 1 is easy, since we can calculate x(1) and show that x(1) > x(0).
Step 2 allows us the assumption that x(k) > x(k-1) > ... > x(0) = 4 > 3
Step 3, the inductive step, simply requires doing what you've already done, but then using Step 2 as justification for x(n) > x(0).
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As for part 2, it is sufficient to show that
for