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

Verified step by step guidance

1. Identify the given initial term: $a_1=3$.
2. Understand the recursive formula: $a_n=4a_{n-1}$ for $n\ge 2$.
3. Calculate the second term using the recursive formula: $a_2=4a_1$.
4. Calculate the third term using the recursive formula: $a_3=4a_2$.
5. Calculate the fourth term using the recursive formula: $a_4=4a_3$.

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 each term based on the preceding one.

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 = 3, which serves as the foundation for calculating subsequent terms. Recognizing the base case is crucial because it provides the first term needed to apply the recursive formula and ensures the sequence has a defined starting point.

Exponential Growth

Exponential growth occurs when a quantity increases by a constant factor over equal intervals. In this sequence, each term is four times the previous term, indicating exponential growth. Understanding this concept helps in predicting the behavior of the sequence as it progresses, as the terms will increase rapidly due to the multiplicative factor.