Recursive formula proofs (HSC SSCE Mathematics Extension 2): Flashcards
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What is a recursive formula in sequences?
What is a recursive formula in sequences?
Defines nth term using previous terms for evaluation.
What does a closed-form definition provide?
What does a closed-form definition provide?
Direct equation for nth term without earlier terms.
What is mathematical induction used for?
What is mathematical induction used for?
To prove statements true for all natural numbers.
What is the base case in induction?
What is the base case in induction?
First step showing the formula works for the first term.
What is the inductive step in induction?
What is the inductive step in induction?
Proves formula holds for n=k+1 using n=k assumption.
How do recursive formulas help in algorithms?
How do recursive formulas help in algorithms?
They define terms using previous terms, aiding complexity.
What is a limitation of recursive definitions?
What is a limitation of recursive definitions?
Can be slow for long sequences due to step-by-step.
What is an advantage of closed-form definitions?
What is an advantage of closed-form definitions?
Efficiently computes terms with direct equations.
What is the Fibonacci sequence's recursive formula?
What is the Fibonacci sequence's recursive formula?
F_n = F_{n-1} + F_{n-2}, with F_0 = 0, F_1 = 1.
What is essential for proving recursive formulas?
What is essential for proving recursive formulas?
Base case and inductive step must be logically followed.