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

Verified step by step guidance

1. Identify the first term of the sequence, which is given as \(a_1 = 4\).
2. Use the recursive formula \(a_n = 2a_{n-1} + 3\) to find the second term by substituting \(n=2\): calculate \(a_2 = 2a_1 + 3\).
3. Find the third term by substituting \(n=3\) into the recursive formula: calculate \(a_3 = 2a_2 + 3\).
4. Find the fourth term by substituting \(n=4\) into the recursive formula: calculate \(a_4 = 2a_3 + 3\).
5. List the first four terms as \(a_1\), \(a_2\), \(a_3\), and \(a_4\) after calculating each term step-by-step using the recursive formula.

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 in order, starting from the initial term, to determine the next terms.

Arithmetic Operations and Order of Evaluation

Accurate calculation of each term requires careful application of arithmetic operations like multiplication and addition, following the correct order of operations. This ensures the sequence values are computed correctly at each step.