The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=7 and a_n=a_n-1 + 5 for n≥2

Verified step by step guidance

1. Identify the initial term of the sequence: \(a_1 = 7\).
2. Recognize the recursive formula: \(a_n = a_{n-1} + 5\) for \(n \geq 2\).
3. Calculate the second term using the recursive formula: \(a_2 = a_1 + 5\).
4. Calculate the third term using the recursive formula: \(a_3 = a_2 + 5\).
5. Calculate the fourth term using the recursive formula: \(a_4 = a_3 + 5\).

Key Concepts

Recursion

Recursion is a method of defining sequences where each term is derived from previous terms. In this case, the first term is given, and subsequent terms are calculated using a specific formula. Understanding recursion is essential for generating terms in sequences, as it allows for the systematic building of terms based on established rules.

Base Case

The base case in a recursive sequence is the initial term or terms from which the sequence begins. For the given sequence, the base case is a_1 = 7. Recognizing the base case is crucial because it provides the starting point for calculating all subsequent terms in the sequence.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. In this example, the difference is 5, as indicated by the formula a_n = a_n-1 + 5. Identifying the nature of the sequence helps in predicting future terms and understanding its overall behavior.