The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a₁=7 and a_n=a_n-1 + 5 for n≥2

Verified step by step guidance

1. Identify the first term of the sequence, which is given as \(a_1 = 7\).
2. Understand the recursive formula: for each term \(a_n\) where \(n \geq 2\), the term is defined as \(a_n = a_{n-1} + 5\). This means each term is 5 more than the previous term.
3. Calculate the second term \(a_2\) by substituting \(n=2\) into the recursive formula: \(a_2 = a_1 + 5\).
4. Calculate the third term \(a_3\) by substituting \(n=3\): \(a_3 = a_2 + 5\).
5. Calculate the fourth term \(a_4\) by substituting \(n=4\): \(a_4 = a_3 + 5\).

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.

Initial Term (Base Case)

The initial term, often denoted as a₁, serves as the starting point of the sequence. It is necessary to know this value to begin applying the recursive formula and calculate the following terms.

Iteration of the Recursive Formula

To find terms beyond the first, repeatedly apply the recursive formula by substituting the previous term's value. This step-by-step process allows you to generate the sequence terms one by one.