What really? The Fibonacci thing is not in the HSC course? It is in the Year 12 2u Grove Textbook in Chapter 7 - Series and my teacher showed us the pattern but not a lot of examples.
May I still have the working out because my teacher assigned that bit for homework. I will clarify with my teacher whether we will be tested on that.
Ok I tracked down the question in the textbook and it looks like it's a
puzzle. (And it got tagged with a hilarious "This is a hard one!" in the MX1 version at least.)
I would be extremely concerned if this appeared in any HSC exam. Nevertheless, here's the full recursive approach.
\[ \text{Let }T_n\text{ be the amount of apples the man has left}\\ \textbf{after }\text{giving the }n\text{-th guard his apples.} \]
Note that the question can be done using 'before' instead of 'after'. You may like to experiment using 'before' instead.
\[ \text{We're essentially given that }T_7 = 1.\\ \text{Our aim is to backtrack to find }T_0\\ \text{which is the amount before even the 1st guard got his apples.} \]
\[ \text{The recurrence of interest is}\\ \boxed{T_n = \frac12 T_{n-1} - 1}.\\ \text{This results from the take-half, then give one more apple scenario}\\ \text{the question presented.} \]
\[ \text{The recurrence relation can be rearranged to}\\ \boxed{T_{n-1} = 2(T_n + 1)}.\\ \text{Once we're here, we can start backtracking.}\\ \begin{align*} T_6 &= 2(1 + 1) = 4\\ T_5 &= 2(4+1) = 10\\ T_4 &= 2(10+1) = 22\\ T_3 &= 2(22+1) = 46\\ T_2 &= 2(46+1) = 94\\ T_1 &= 2(94+1) = 190\\ T_0 &= 2(190+1) = 382\end{align*} \]
\[ \text{So he originally had 382 apples.} \]
Note that the problem was put in the geometric series section, so it had to somehow be related to geometric series. It turns out that one can actually prove that \(T_n = 384(2^{-n}) - 2\). This looks
almost like a geometric series, except there is an extra \(-2\) hanging on the end interfering with the pattern. The formal derivation more or less uses first year uni techniques and is hence avoided, but there may be an intuition behind this formula that I haven't had the time to investigate yet.