Problem 12 - Algebra 2

Question

Write a recursive formula for each sequence. Then find the next term. $$ -2,-1,0,1,2, \ldots $$

Step-by-Step Solution

Answer

The recursive formula for the given sequence is \(a_{n} = a_{n-1} + 1\). The next term in the sequence is 3.

Step 1: Identify the Common Difference

The common difference d in an arithmetic sequence can be found by subtracting any term from the term that follows it. Here, if we subtract -2 (the first term) from -1 (the second term), we get a difference of 1. Hence, d=1.

Step 2: Write the Recursive Formula

The recursive formula for an arithmetic sequence can be written as a_(n) = a_(n-1) + d. Here, since d=1, our recursive formula becomes a_(n) = a_(n-1) + 1.

Step 3: Find the Next Term

We can use our recursive formula a_(n) = a_(n-1) + 1 to find the next term in the sequence. Subtracting one from the last term given in the sequence (which is 2) and adding one to the result gives us 3. Hence, the next term in the sequence is 3.

Key Concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term. This type of sequence is straightforward and easy to recognize due to the consistent difference between each pair of consecutive terms.

In an arithmetic sequence like \(-2, -1, 0, 1, 2, \ldots\) each number can be derived by simply adding the same number to the previous number. This pattern is what defines the arithmetic nature of the sequence.

• Each number in the sequence is called a "term."
• The difference added to move from one term to the next is the "common difference."

A good way to think of an arithmetic sequence is to visualize it as a step-by-step movement across a number line, where each step is equal in size.

Exploring the Common Difference

The common difference is a crucial part of an arithmetic sequence. It is the fixed amount added to each term to get to the next term. Identifying this difference allows you to predict further terms in the sequence.

In the sequence \(-2, -1, 0, 1, 2, \ldots\), the common difference is 1. This is because each term is exactly 1 more than the term before it. To find the common difference, you can pick any term, say \(-1\), and subtract the term that comes just before it, \(-2\):

• -1 - (-2) = 1

Once the common difference is known, it explains how the sequence expands over time. Each time you want to find a subsequent term; simply add this value to the previous term.

Identifying Terms in a Sequence

Terms in a sequence are the individual numbers that make up the sequence. In arithmetic sequences, these terms have a special arrangement formed by repeatedly adding the common difference to the initial term.

Let's consider the sequence \(-2, -1, 0, 1, 2, \ldots\). To understand how the terms are connected, look at the first term \(-2\). By adding the common difference of 1, you get the next term \(-1\), and continuing this process, you can list further terms.

• First term: -2
• Next term: -1 (add 1)
• Following term: 0 (add 1)

Using the recursive formula \(a_{n} = a_{n-1} + 1\), you can find any term in the sequence if you know the previous term. For example, if the last term you know is 2, then \(2 + 1 = 3\), and so on. This simple process helps in building and understanding the entire sequence effortlessly.