Fibonacci Sequence

Context: FIT1058_MOC, FIT2004_MOC · a two-term recurrence with no obvious closed form · bounded by Mathematical Induction · closed form built from a quadratic’s roots FIT2004 use: §4 and the Proof Blueprint (Applied 2 P7) — the matrix identity and the doubling identities that turn the naive $O(2^n)$ recursion into a halving one ➔ Analysing Recursive Algorithms (Time and Auxiliary Space)

Quick Revision

• 🎯 Objective: $f_1=f_2=1,f_n=f_{n-1}+f_{n-2}$ ➔ $1,1,2,3,5,8,13,\dots$.
• 📦 Core Components: two base cases ➔ strong induction bounds ➔ Binet closed form.
• ⚡ Key Constraint: $f_n=\Theta (r_1^n)$, $r_1=\frac{1+\sqrt{5}}{2}$; exact formula needs both roots of $r^2=r+1$.

📝 Core

1. The Recurrence

• Definition ➔ $f_1=f_2=1$, $f_n=f_{n-1}+f_{n-2}$ ($n\ge 3$).
• Two base cases ➔ the rule only applies for $n\ge 3$.
• Standard example ➔ bounding a sequence before/instead of an exact formula.

2. Bounding by Induction

• Upper ➔ $f_n\le 2^n$ (basis $f_1,f_2$; step $2^k+2^{k-1}<2^{k+1}$).
• Lower ➔ $f_n\ge \frac{4}{9}\cdot 1.5^n$ ($\frac{4}{9}$ chosen so both bases hold).
• Sharp ➔ $r_1^{n-2}\le f_n\le r_1^n$, $r_1=\frac{1+\sqrt{5}}{2}$.

3. Exact (Binet) Formula

• Characteristic ➔ roots of $r^2-r-1=0$: $r_1=\frac{1+\sqrt{5}}{2}$, $r_2=\frac{1-\sqrt{5}}{2}$.
• Form ➔ $f_n={\alpha }_1{r}_1^n+{\alpha }_2{r}_2^n$, matched by ${\alpha }_1=-{\alpha }_2=\frac{1}{\sqrt{5}}$.

Key identities:

$f_n\le 2^n\text{ (induction)},r^2=r+1\Rightarrow r_1=\frac{1+\sqrt{5}}{2},r_2=\frac{1-\sqrt{5}}{2}$ $f_n=\frac{1}{\sqrt{5}}{\left(\frac{1+\sqrt{5}}{2}\right)}^n-\frac{1}{\sqrt{5}}{\left(\frac{1-\sqrt{5}}{2}\right)}^n\text{ (Binet)}$

4. Matrix Form and the Doubling Identities (FIT2004)

• ⚠️ Indexing shifts here ➔ FIT2004 uses $F(0)=0$, $F(1)=1$; FIT1058 starts at $f_1=f_2=1$. The two agree for every $n\ge 1$ — only the extra $F(0)=0$ is new — but a proof written in the wrong convention fails its own base case.
• The matrix identity ➔ one matrix power encodes two consecutive Fibonacci numbers:

$${\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]}^n=\left[\begin{array}{cc}F(n+1)& F(n)\\ F(n)& F(n-1)\end{array}\right],n\ge 1$$

• Doubling falls straight out of $M^{2k}=(M^k{)}^2$ ➔ squaring the right-hand side and reading off entries gives

$$\begin{array}{rlrl}F(2k+1)& =F(k+1{)}^2+F(k{)}^2& & \text{(top-left entry)}\\ F(2k)& =F(k)F(k+1)+F(k-1)F(k)& & \text{(top-right entry)}\\ & =F(k)[2F(k+1)-F(k)]& & \text{(substitute }F(k-1)=F(k+1)-F(k))\end{array}$$

• The elimination step is the marked one ➔ the required identity contains no $F(k-1)$, so rearranging the defining recurrence into $F(k-1)=F(k+1)-F(k)$ and substituting is what closes the derivation.
• Why an algorithms unit cares ➔ both identities express index $2k$ in terms of index $k$, so evaluating them halves the index each step ⟹ recursion depth $\Theta (\log n)$ against the naive recurrence’s $\Theta (n)$ depth and $O(2^n)$ time. Same move as power_fast: divide the parameter instead of decrementing it.
• $F(k)$ appears twice in each identity ➔ compute it once and reuse it, or the halving is spent on a binary call tree and the win evaporates ➔ the duplicate-call trap in Analysing Recursive Algorithms (Time and Auxiliary Space).

🧮 Proof Blueprint — the matrix identity by induction

Theorem. $L(n)=R(n)$ for all $n\ge 1$, where $L(n)={\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]}^n$ and $R(n)=\left[\begin{array}{cc}F(n+1)& F(n)\\ F(n)& F(n-1)\end{array}\right]$. Strategy. Induction on $n$; peel one factor off the power, invoke the hypothesis, and let the defining recurrence re-fold the entries.

Base case ($n=1$):

$$L(1)=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}F(2)& F(1)\\ F(1)& F(0)\end{array}\right]=R(1)✓$$

Inductive step. Assume $L(k)=R(k)$ for some $k\ge 1$. Then

$$\begin{array}{rlrl}L(k+1)& ={\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]}^k\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]=L(k)\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]& & \\ & =R(k)\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]& & \text{(inductive hypothesis)}\\ & =\left[\begin{array}{cc}F(k+1)+F(k)& F(k+1)\\ F(k)+F(k-1)& F(k)\end{array}\right]& & \\ & =\left[\begin{array}{cc}F(k+2)& F(k+1)\\ F(k+1)& F(k)\end{array}\right]=R(k+1)& & ■\end{array}$$

• The whole proof is one factorisation ➔ writing $L(k+1)=L(k)\cdot L(1)$ is what creates a place for the hypothesis to be used; students who expand $L(k+1)$ directly have nothing to substitute into.
• The recurrence does the re-folding ➔ every entry of the product is a sum of two consecutive Fibonacci numbers, so $F(a)+F(a-1)=F(a+1)$ converts the product back into $R$-shape. No arithmetic beyond that is needed.

⚖️ Core Decision Matrix

Goal	Roots used	Base cases matched
upper bound $r_1^n$	$r_1$ only	one (inequality)
lower bound	$r_1$ only	one
exact Binet	$r_1$ and $r_2$	two (equality)
growth	$r_1$	$\Theta (r_1^n)$

When It Flips: a bound of form $r^n$ needs $r^2\ge r+1$ (one free constant, one base case); an exact formula needs equality $r^2=r+1$ and both roots (two constants, two base cases). $r_2<0$ can't be a bound but is essential in Binet. Ratio $f_{n+1}/f_n\to r_1$ (golden ratio).

📊 Exam Execution Trace

Applied Exercise

Problem: Compute $f_1..f_7$ and verify Binet at $n=7$. Derivation Proof / Hand-Calculation Walkthrough:

$$\begin{array}{rl}{f}_1..f_7& =1,1,2,3,5,8,13\\ f_7& =\frac{1}{\sqrt{5}}(r_1^7-r_2^7)=\frac{1}{\sqrt{5}}(29.03\dots +0.03\dots )=13✓\end{array}$$

Final Extracted Output: $f_7=13$ from both recurrence and Binet.

⚠️ Common Mistakes

• 💡 Two base cases required ➔ the step uses the hypothesis at $k$ and $k-1$ (strong induction), so $n=1,2$ must be checked separately.
• 💡 Mixing the two indexing conventions ➔ the matrix identity’s base case reads $F(0)=0$ in the bottom-right corner; carrying FIT1058’s $f_1=f_2=1$ into it makes $R(1)$ wrong and the induction unprovable.
• 💡 Expanding $L(k+1)$ instead of factorising it ➔ the inductive hypothesis can only be invoked once $L(k)$ appears literally, so the step must start from $L(k+1)=L(k)\cdot L(1)$.

🧠 Active Recall

Why does proving $f_n\le 2^n$ require two base cases?

• Hint: Strong induction.

Answer

• Short answer: The rule holds only for $n\ge 3$; $f_1,f_2$ checked directly, and the step uses $k$ and $k-1$.
• Why: Two predecessors ➔ needs two consecutive base cases.

The doubling identities express $F(2k)$ and $F(2k+1)$ via $F(k)$ and $F(k+1)$. Why is that an algorithmic result, not just an algebraic curiosity?

• Hint: Compare how the index moves against the naive recurrence.

Answer

• Short answer: the naive recurrence decrements the index and branches, giving depth $\Theta (n)$ and $O(2^n)$ time; the doubling identities halve it, so the depth drops to $\Theta (\log n)$.
• Why: Divide the parameter, don’t decrement it ➔ this is the same shift that separates power from power_fast, and it only pays off if the repeated $F(k)$ term is computed once and reused — otherwise the halved depth is refilled by a binary call tree ➔ Analysing Recursive Algorithms (Time and Auxiliary Space).

Why do bounds need only one root but Binet needs both?

• Hint: Inequality vs equality.

Answer

• Short answer: A bound $r^n$ has one free constant (one base case, inequality); an exact formula matches two base cases with equality, needing both roots.
• Why: $r_2<0$ ➔ breaks the basis as a bound but is indispensable in Binet.