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To understand a sequence formula in the context of recurrence relations, one must first grasp how each term in a sequence is defined. This involves using a base term and a recurrence relation. In this problem, we start with the first term, \( a_1 = \frac{3}{2} \), and use the recurrence relationship \( a_n = 5a_{n-1} - 1 \) to generate further terms. By systematically applying this formula, we calculated the sequence's first six terms:

• \( a_1 = \frac{3}{2} \)
• \( a_2 = \frac{13}{2} \)
• \( a_3 = \frac{63}{2} \)
• \( a_4 = \frac{313}{2} \)
• \( a_5 = \frac{1563}{2} \)
• \( a_6 = \frac{7813}{2} \)

The goal was to discover a formula that encapsulates the behavior of the sequence in relation to \( n \). By observing these terms, it becomes evident they follow a pattern, specifically resembling the sequence formula \( \frac{5^n - 1}{2} \). This formula elegantly summarizes what the recurrence relation does with each step.

Mathematical induction is a technique used to prove hypotheses about formulas or sequences. It is structured as a two-step logic process: the base case, and the inductive step. Here, it's used to verify the guessed sequence formula, \( a_n = \frac{5^n - 1}{2} \), accurately represents any term in the sequence.**Base Case:**
Start with the first term: we observe that \( a_1 = \frac{3}{2} \), aligning perfectly with our formula since \( \frac{5^1 - 1}{2} = \frac{3}{2} \). **Inductive Step:**
Assume it holds for \( n = k \), that is, \( a_k = \frac{5^k - 1}{2} \). For \( n = k+1 \), test the relation:

• Apply the recurrence relation \( a_{k+1} = 5a_k - 1 \)
• Substitute the formula for \( a_k \): \( 5 \left( \frac{5^k - 1}{2} \right) - 1 \)
• Transform this expression to match our conjecture: \( \frac{5^{k+1} - 1}{2} \)

Both cases confirm the formula via induction, completing the proof.

Recursive sequences rely on defining each term based on prior terms in the sequence, typically through a recurrence relation. They provide an approach to define complex patterns incrementally. In this specific example, the relation \( a_n = 5a_{n-1} - 1 \) shows how each element emerges from its predecessor by multiplying it by 5 and then subtracting 1.**Properties of Recursive Sequences:**

• **Initial Condition:** Establishes the starting point, which is essential for generating any further terms.
• **Recurrence Relation:** Guides the transformation from one term to the next.
• **Predictability:** Once understood, the sequence can be generated completely from the initial condition and the relation applied successively.
• **Application:** Useful in solving a vast array of problems, from computer algorithms to economic forecasting.

These elements together generate a sequence, showcasing the power and flexibility of recursive definitions in mathematics.