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Mathematical induction is a powerful technique often used to prove that a statement, formula, or property holds for all natural numbers. The process resembles a chain reaction, where the validity at step one ensures the validity at the next step and so on.

There are two primary phases in applying mathematical induction:

• Base Case: First, we demonstrate that the statement holds true for the initial value, often denoted as \(n = 1\). In the given sequence \(s_n\), checking the initial condition \(s_1 = 1\) ensures that our base case is valid.
• Inductive Step: Assume that the statement is true for some number \(k\), and then prove it for \(k + 1\). In the exercise, assuming \(s_k = 3k - 2\) and showing it holds for \(s_{k+1}\) establishes the inductive step.

Once both steps are verified, the entire range of natural numbers can be covered by these two connections, ensuring the mathematical statement's truth across all values.

A sequence is an ordered list of numbers defined by a specific rule. In mathematics, sequences are one of the fundamental concepts as they allow modeling and solving various problems.

The sequence \(s_n = 3n - 2\) is said to be defined explicitly, meaning each term can be calculated directly without relying on the previous one.
The explicit formula gives us a more straightforward calculation:

• For \(n = 1\), as shown: \(s_1 = 3(1) - 2 = 1\).
• For \(n = k + 1\), \(s_{k+1} = 3(k + 1) - 2 = 3k + 3 - 2 = 3k + 1\).

Sequences can also be expressed recursively, with each term constructed from its predecessors. In the given problem, this form is \(s_{n} = s_{n-1} + 3\), which shows the sequential addition of 3 to generate the next term. Understanding both explicit and recursive forms is crucial to analyzing relationships and patterns in sequences.

An initial condition is the starting point for a sequence or recurrence relation. It specifies the value of the first term, laying the foundation for further calculations.

For recurrence relations, the initial condition acts as the anchor point:

• In the context of the sequence \(s_n\), the initial condition is \(s_1 = 1\).
• It helps to uniquely determine the terms that follow, based on the rule defined by the recurrence relation \(s_{n} = s_{n-1} + 3\).

The significance of an initial condition lies in ensuring that the recursive formula produces the exact sequence expected. Without the correct starting point, even a properly defined recurrence relation can yield incorrect results. That's why the verification of the initial condition \(s_1 = 1\) is a crucial step, ensuring the formula's applicability to generate subsequent terms accurately.