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The Fibonacci sequence is a famous series of numbers that starts with 0 and 1, and each subsequent number is the sum of the two preceding ones. Here's how it begins:
\[ F_0 = 0, \; F_1 = 1, \; F_2 = F_0 + F_1 = 1, \; F_3 = F_1 + F_2 = 2, \; \text{and so on.} \]
This sequence appears in many different areas of mathematics and science, such as biology, architecture, and partial differential equations. It is often used to model growth patterns like flower petals or shell spirals.

• **Key Properties:** The defining rule of Fibonacci numbers is \( F_{n} = F_{n-1} + F_{n-2} \).
• **Indexing:** By convention, the sequence starts with an index of 0.
• **Recurrent Relationships:** The Fibonacci numbers are used in various forms of recurrence relations, a common topic in mathematics.

In the given exercise, Fibonacci numbers are directly used in the equation \( e_{n}=2 F_{n}-2 \). By understanding this sequence and its properties, students can better grasp how values evolve in a recursive manner.

The basis of induction, also known as the base case, is the starting point of a proof by induction. It's like the first domino in a line of dominoes—once it falls, the rest follow under the right conditions.
In a mathematical induction proof, the base case demonstrates that a property holds for the initial value. This ensures that the pattern kicks off correctly.

• **Verification:** We start by confirming the statement holds true for \( n = 0 \) and sometimes \( n = 1 \).
• **Exercise Example:** In the exercise provided, this involved showing \( e_0 = 2 F_0 - 2 = -2 \) and \( e_1 = 2 F_1 - 2 = 0 \).

Establishing this basis is critical because it provides a solid foundation, ensuring that any properties assumed in further steps are starting from a true point. Without this, any subsequent steps in an induction proof can't be guaranteed to hold true.

The inductive step is a pivotal part of mathematical induction. It acts as a bridge that carries the truth from one instance to the next. To successfully carry out this step in a proof involves two main ideas.
First, the inductive hypothesis, where you assume the statement is true for some arbitrary case, say \( n = k \). In our exercise, we assume that \( e_k = 2F_k - 2 \) is true.
Second, using the inductive hypothesis to prove the next case, \( n = k+1 \). Essentially, you show that if the statement holds for \( n = k \), it must hold for \( n = k+1 \) as well.

• **Transitive Property:** By proving one case leads to another, an infinite number of cases can be concluded to be true.
• **Exercise Context:** Here, the proof required demonstrating that \( e_{k+1} = 2F_{k+1} - 2 \) follows from the assumption that \( e_k = 2F_k - 2 \) and \( e_{k-1} = 2F_{k-1} - 2 \).

Successful completion of the inductive step, combined with a true base case, seals the induction proof by showing the statement is true for all natural numbers.