Problem 21 - A Graphical Approach to College Algebra

Question

Find the first four terms of each sequence. $$a_{1}=-2, a_{n}=a_{n-1}+3, \text { for } n>1$$

Step-by-Step Solution

Answer

The first four terms are -2, 1, 4, and 7.

Step 1: Understand the Given Information

The sequence starts with the first term, \( a_1 = -2 \). Each subsequent term is given by the recursive formula \( a_n = a_{n-1} + 3 \). This means each term is 3 more than the previous term.

Step 2: Compute the First Term

The first term is already provided: \( a_1 = -2 \).

Step 3: Calculate the Second Term

Using the formula, \( a_2 = a_1 + 3 = -2 + 3 = 1 \).

Step 4: Determine the Third Term

Again, use the formula to find the next term: \( a_3 = a_2 + 3 = 1 + 3 = 4 \).

Step 5: Find the Fourth Term

Continuing the pattern, the fourth term: \( a_4 = a_3 + 3 = 4 + 3 = 7 \).

Key Concepts

Recursive Formula

A recursive formula is a mathematical expression that defines each term in a sequence using the preceding terms. In simple terms, it tells you how to get from one term to the next.
In this exercise, the recursive formula given is \( a_n = a_{n-1} + 3 \). This means that to get any term \( a_n \), you simply take the term before it, \( a_{n-1} \), and add 3.
This formula is crucial because it defines the entire sequence step by step, showing how each term builds from the last. By knowing the first term, in this case, \( a_1 = -2 \), and the recursive pattern, you can find any term in the sequence. The key benefit of a recursive formula is its simplicity in defining complex sequences and its ability to provide a structured way to calculate terms.

Arithmetic Progression

An arithmetic progression (AP), often referred to as an arithmetic sequence, is a sequence of numbers where each term after the first is obtained by adding a constant difference to the previous term. This constant is known as the common difference.
In the exercise sequence, the common difference is 3, as each term is 3 more than the preceding term. This is a hallmark of an arithmetic progression.
To identify an arithmetic progression, look for uniform spacing between terms. This spacing is what defines the sequence's structure and progression. Understanding this characteristic allows you to predict future terms and understand the overall trend of the sequence.
For example: if you start with \( a_1 = -2 \) and consistently add 3, you get \( -2, 1, 4, 7, \ldots \), displaying the clear pattern of an arithmetic progression.

Sequence Terms

Sequence terms refer to the individual elements that make up a sequence. Each term is typically a number, and together the terms form a series. In any given sequence, each term is labeled in an ordered manner, usually by subscript numbers indicating their position.
In the exercise, we calculated the first four terms as follows:

• The first term is \( a_1 = -2 \).
• The second term is \( a_2 = 1 \).
• The third term is \( a_3 = 4 \).
• The fourth term is \( a_4 = 7 \).

Each term is derived from the formula and shows a progression where each subsequent term is 3 units greater than the one before.
Understanding sequence terms is crucial because it allows us to comprehend patterns, make predictions, and solve complex problems in mathematics involving sequences.