The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=4 and a_n=2a_n-1 + 3 for n≥2

Verified step by step guidance

1. Start by identifying 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 \( a_2 \). Substitute \( a_1 \) into the formula: \( a_2 = 2 \times 4 + 3 \).
3. Calculate the third term \( a_3 \) using the recursive formula. Substitute \( a_2 \) into the formula: \( a_3 = 2a_2 + 3 \).
4. Find the fourth term \( a_4 \) by substituting \( a_3 \) into the recursive formula: \( a_4 = 2a_3 + 3 \).
5. List the first four terms of the sequence: \( a_1, a_2, a_3, a_4 \).

Key Concepts

Recursion

Recursion is a method of defining sequences or functions where the next term is derived from previous terms. In this case, the sequence is defined by an initial term and a recursive formula that relates each term to its predecessor. Understanding recursion is essential for generating terms in sequences, as it allows for the systematic calculation of terms based on established rules.

Base Case

The base case is the initial condition or starting point of a recursive sequence. In the given problem, the base case is defined as a_1 = 4, which serves as the foundation for calculating subsequent terms. Recognizing the base case is crucial because it provides the first value needed to apply the recursive formula and ensures the sequence has a defined starting point.

Recursive Formula

A recursive formula expresses each term of a sequence in relation to one or more previous terms. In this question, the recursive formula is a_n = 2a_n-1 + 3, which indicates how to compute each term based on the preceding term. Understanding how to manipulate and apply recursive formulas is vital for generating the terms of the sequence accurately.