Final answer:
Proof of the Fibonacci sequence via induction starts with the established base case for the first two numbers in the series, then proceeds to the induction step that applies the definitional recurrence to show the pattern holds for subsequent numbers.
Step-by-step explanation:
The Fibonacci sequence can be proven by mathematical induction which is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers. It consists of two steps: the base case and the induction step.
To prove the Fibonacci sequence by induction, first establish the base case for the first two numbers in the sequence, which are 1 (or sometimes 0) and 1. These are given to be true by definition. Then assume the nth and (n-1)th terms are F(n) and F(n-1), respectively, and that they fit the Fibonacci sequence, where F(n) = F(n-1) + F(n-2). Next, you must show that F(n+1) = F(n) + F(n-1) also fits this pattern, using the assumption for F(n) and F(n-1). This establishes the induction step.