Problem 43 - Beginning and Intermediate Algebra

Question

Determine whether each sequence is arithmetic or geometric. Then, find the general term, \(a_{m}\), of the sequence. $$-31 .-24-17,-10,-3, \ldots$$

Step-by-Step Solution

Answer

The given sequence is an arithmetic sequence with a common difference of \(d = 7\). The general term of this sequence can be expressed as: \(a_{m} = -31 + (m - 1)7\).

Step 1: Determine if the sequence is arithmetic or geometric

We will find the differences and the ratios between consecutive terms. The sequence is given as: $$-31, -24, -17, -10, -3, \ldots$$ Differences between terms: \((-24 - (-31)), (-17 - (-24)), (-10 - (-17)), (-3 - (-10))\) By calculating the differences, we obtain \(7, 7, 7, 7\). All the differences are constant, so it's an arithmetic sequence. Now let's find the common difference, \(d\): $$d = -24 - (-31) = 7$$

Step 2: Find the general term of the arithmetic sequence

The general term of an arithmetic sequence is given by the formula: $$a_{m} = a_{1} + (m - 1)d$$ Where \(a_{m}\) is the m-th term, \(a_{1}\) is the first term, \(m\) is the number of terms, and \(d\) is the common difference. In this sequence, \(a_{1} = -31\), and \(d = 7\). So, the general term of the sequence is: $$a_{m} = -31 + (m - 1)7$$

Key Concepts

General Term

In an arithmetic sequence, the **general term** plays a crucial role in understanding the sequence's structure. It's essentially a formula that allows you to find any term in the sequence without listing all the previous terms. For arithmetic sequences, the general term is determined using the formula:
\[ a_m = a_1 + (m - 1) d \]where:

• \(a_m\) is the m-th term you're interested in
• \(a_1\) is the first term of the sequence
• \(m\) is the term number
• \(d\) is the common difference between terms

Let's take our example sequence:

• First term, \(a_1 = -31\)
• Common difference, \(d = 7\)

By substituting these values into the general term formula, we get: \[ a_m = -31 + (m - 1) \times 7 \]With this formula, you can quickly compute any term in the sequence without needing to write the complete list.

Common Difference

The **common difference** is a key feature of an arithmetic sequence. It's the constant number you add (or subtract if negative) to any term to get the next term. To identify it, you calculate the difference between any pair of successive terms in the sequence.In our example, the sequence provided is:
\(-31, -24, -17, -10, -3, \ldots\)
Calculate the differences:

• \(-24 - (-31) = 7\)
• \(-17 - (-24) = 7\)
• \(-10 - (-17) = 7\)
• \(-3 - (-10) = 7\)

Each time, the difference is \(7\). This tells us that the sequence increases by the same amount each step, confirming it's an arithmetic sequence. This steady increase is what makes finding terms with the general formula practical and straightforward.

Sequence Type Determination

Before diving into specific formulas, it's essential to understand the **sequence type** we're dealing with. There are primarily two types of sequences: arithmetic and geometric. Recognizing the sequence type helps in using the right approach to solve problems.To determine the type of a sequence:

• In an **arithmetic sequence**, each term increases or decreases by a consistent amount. Checking differences between successive terms will reveal this consistency.
• In a **geometric sequence**, each term is multiplied by a constant factor to get the next term. Compare the ratio of successive terms; if it remains constant, it's geometric.

For the sequence \(-31, -24, -17, -10, -3, \ldots\), we calculated the consecutive differences all as \(7\), indicating an arithmetic sequence. Understanding these characteristics enables you to efficiently apply the appropriate mathematical formulas to find specific terms or understand the sequence's behavior over time.