Problem 22 - A Graphical Approach to College Algebra

Question

Find the first four terms of each sequence. $$a_{1}=-1, a_{n}=a_{n-1}-4, \text { for } n>1$$

Step-by-Step Solution

Answer

The first four terms are -1, -5, -9, and -13.

Step 1: Understand the sequence formula

The sequence is given by the initial term \( a_1 = -1 \) and the recursive formula \( a_n = a_{n-1} - 4 \) for \( n > 1 \). This means each term is obtained by subtracting 4 from the previous term.

Step 2: Find the first term of the sequence

The first term is given directly as \( a_1 = -1 \).

Step 3: Calculate the second term

Using the recursive formula, calculate the second term: \( a_2 = a_1 - 4 = -1 - 4 = -5 \).

Step 4: Calculate the third term

Use the recursive formula to find the third term: \( a_3 = a_2 - 4 = -5 - 4 = -9 \).

Step 5: Calculate the fourth term

Again, apply the recursive formula to find the fourth term: \( a_4 = a_3 - 4 = -9 - 4 = -13 \).

Key Concepts

Sequence Formula

In mathematics, sequences are ordered lists of numbers where the order and pattern of numbers follow a specific logic. The sequence formula is a way to describe how to find any term in a sequence.

• There are different types of sequences, such as arithmetic, geometric, and more. Each has its own distinctive formulas.
• In the given exercise, the sequence is described by the formula for the first term, \( a_1 = -1 \), and a recursive relationship for subsequent terms.

The formula \( a_n = a_{n-1} - 4 \) is used to find subsequent terms after the initial term. This means to find any term \( a_n \) in this sequence, you would subtract 4 from the previous term \( a_{n-1} \). This formula indicates an arithmetic sequence where each term decreases by 4 from the prior term.

Recursive Formula

A recursive formula specifies each term in a sequence using the previous term(s). This is a powerful method, particularly for sequences where each term relates closely to its predecessors.

• The exercise provides us with a recursive formula: \( a_n = a_{n-1} - 4 \), applied for \( n > 1 \).
• Recursive formulas require you to know the initial term to generate future terms. Here, \( a_1 = -1 \) serves as the starting point.

To practically use a recursive formula, after identifying \( a_1 \), proceed by using the formula to calculate \( a_2 \), then \( a_3 \), and continue similarly. This approach processes the sequence step-by-step, allowing you to determine each subsequent term as shown in this exercise.

Sequence Terms

Sequence terms are the individual numbers in a sequence, each labeled with an index. The index helps in identifying the position of the term in the sequence (e.g., \( a_1 \), \( a_2 \), \( a_3 \), etc.).

• The first term or initial term is explicitly given in this exercise, \( a_1 = -1 \).
• Using the recursive formula, further terms can be calculated one by one.
• In this sequence, the next terms are derived by continually subtracting 4 from the previous term.

For example, the second term \( a_2 \) is \( -5 \), the third term \( a_3 \) is \( -9 \), and the fourth term \( a_4 \) is \( -13 \). In any sequence, each term builds upon the information or values provided by preceding terms, highlighting the connected nature of sequence terms.