The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a₁=3 and a_n=4a_n-1 for n≥2

Verified step by step guidance

1. Identify the given recursive formula: the first term is \(a_1 = 3\), and for \(n \geq 2\), the sequence is defined by \(a_n = 4a_{n-1}\).
2. Start by writing down the first term explicitly: \(a_1 = 3\).
3. Find the second term by substituting \(n=2\) into the recursive formula: \(a_2 = 4a_1\).
4. Find the third term by substituting \(n=3\): \(a_3 = 4a_2\).
5. Find the fourth term by substituting \(n=4\): \(a_4 = 4a_3\).

Key Concepts

Recursive Sequence Definition

A recursive sequence is defined by specifying the first term(s) and a formula that relates each term to one or more previous terms. Understanding how to use the given formula to find subsequent terms is essential for generating the sequence.

Substitution Method for Finding Terms

To find terms in a recursive sequence, substitute the previous term(s) into the recursive formula step-by-step. This process involves calculating each term based on the value of the term before it, starting from the initial term.

Notation and Indexing in Sequences

Sequences use subscript notation (aₙ) to denote the nth term. Understanding the indexing, especially the starting point (usually n=1), is crucial to correctly apply the recursive formula and list the terms in order.