Solving Recurrences

Fibonacci Sequence

• Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
• Base Cases:
  • $F_0 = 0$
  • $F_1 = 1$
• Recurrence Relation:
  • $F_n = F_{n-1} + F_{n-2}$ for $n > 2$

Inductive Proof

• Method Used: Strong induction
• Result Proven: $E_n = (\phi^n - (-\phi)^{-n})$

Golden Ratio

• Definition: Often denoted by $\phi$
• Importance: Key in solving the Fibonacci recurrence relation

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