Problem 23 - A Graphical Approach to Precalculus with Limits

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: Identify the given information

We are given the initial term of the sequence, which is \(a_1 = -2\). The recursive formula for the sequence is \(a_n = a_{n-1} + 3\) for \(n > 1\). This means each term is obtained by adding 3 to the previous term.

Step 2: Find the second term

To find \(a_2\), use the recursive formula with \(n = 2\): \(a_2 = a_1 + 3 = -2 + 3 = 1\). So, the second term \(a_2\) is 1.

Step 3: Find the third term

To find \(a_3\), use the recursive formula with \(n = 3\): \(a_3 = a_2 + 3 = 1 + 3 = 4\). Thus, the third term \(a_3\) is 4.

Step 4: Find the fourth term

To find \(a_4\), use the recursive formula with \(n = 4\): \(a_4 = a_3 + 3 = 4 + 3 = 7\). Therefore, the fourth term \(a_4\) is 7.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a type of sequence in mathematics where each term is derived by adding a constant value to the previous term. This constant value is known as the "common difference."
For instance, in the sequence provided by the exercise, the common difference is 3. You start with the initial term, which is -2, and keep adding 3 to get the next terms in the sequence.

• The sequence starts at -2.
• Then, adding 3 leads to the next term, 1.
• Continuing this pattern results in further terms 4, and then 7.

Understanding arithmetic sequences allows you to quickly generate terms in a sequence and recognize the pattern of adding a fixed number.

Recursive Formula

A recursive formula expresses each term of a sequence as a function of the preceding term. It’s a powerful way to describe sequences like the arithmetic ones. In the context of our exercise, the recursive formula given is: \[ a_n = a_{n-1} + 3 \] This formula represents a step-by-step method to find any term, as long as you know the previous term. To start, we need the initial term, also called the seed value, which is provided as \( a_1 = -2 \).

• From there, each subsequent term is the previous term \( a_{n-1} \) plus 3.
• Say you want the third term \( a_3 \); you first calculate the second term \( a_2 \), and so forth.

Recursive formulas are ideal when you want to calculate early sequence terms iteratively, starting from a known initial term.

Sequence Terms

Sequence terms are the individual elements or numbers that make up a sequence. Each term has a position based on the order in which it appears. In our arithmetic sequence example, terms have both a numeric value and a position number (first, second, third, etc.).
To identify terms:

• The first term \( a_1 \) is given as -2.
• The recursive formula helps to find the following terms:
• The second term \( a_2 \) is 1, calculated as \( a_1 + 3 \).
• The third term \( a_3 \) is 4, using \( a_2 + 3 \).
• The fourth term \( a_4 \) is 7, determined by \( a_3 + 3 \).

Understanding sequence terms helps in analyzing the growth pattern. Each term results from the systematic application of the recursive formula.