Problem 37 - Beginning and Intermediate Algebra

Question

Determine whether each sequence is arithmetic or geometric. Then, find the general term, \(a_{m}\), of the sequence. $$15,24,33,42,51, \dots$$

Step-by-Step Solution

Answer

The given sequence is arithmetic with a common difference of 9. The general term of the sequence is \(a_{m} = 6 + 9m\).

Step 1: Identify the sequence type (arithmetic or geometric)

First, we will try to identify if there is a common difference between consecutive terms in the sequence or if there is a common ratio between consecutive terms. To check for an arithmetic sequence, we will find the difference between consecutive terms: $$24-15=9\\ 33-24=9\\42-33=9\\ 51-42 =9$$ All differences between consecutive terms are equal (they are all 9). This indicates that the sequence is arithmetic. Now that we know the sequence is arithmetic, we are going to find the general term, \(a_{m}\).

Step 2: Finding the general term of the arithmetic sequence

To find the general term of an arithmetic sequence, we use the formula: $$a_{m} = a_1 + (m-1)d$$ Where \(a_m\) is the general term, \(a_1\) is the first term, \(m\) is the position of the term in the sequence, and \(d\) is the common difference. In our sequence, we have \(a_1 = 15\) and the common difference \(d=9\). Now, plug these values into the formula: $$a_{m} = 15 + (m-1)9$$

Step 3: Simplify the general term formula

Now, let's simplify the formula: $$a_{m} = 15 + 9m - 9$$ Combine like terms: $$a_{m} = 6 + 9m$$ The general term of the sequence is: $$a_{m} = 6 + 9m$$

Key Concepts

Understanding Common Difference

The common difference is a critical aspect of any arithmetic sequence. It represents the constant interval between each pair of consecutive terms. Finding this difference is the first step in identifying an arithmetic sequence.
For example, in the sequence 15, 24, 33, 42, 51, we calculate the difference between each pair of terms:

• 24 - 15 = 9
• 33 - 24 = 9
• 42 - 33 = 9
• 51 - 42 = 9

All these differences are the same, which confirms that the sequence is arithmetic.
This repeated interval is what characterizes the arithmetic sequence and is fundamental in determining the general formula.

Deriving the General Term Formula

Once we've identified the arithmetic sequence and found the common difference, we can derive the general formula, known as the general term formula. This formula helps find any term in the sequence without listing all preceding terms.
The general term formula for an arithmetic sequence is \[ a_m = a_1 + (m-1)d \] , where:

• \(a_m\) is the \(m^{th}\) term,
• \(a_1\) is the first term,
• \(m\) is the position of the term,
• \(d\) is the common difference.

In our sequence, \(a_1 = 15\) and \(d = 9\). Substituting the known values, we derive the formula: \[ a_m = 15 + (m-1) \times 9 \]
Simplifying gives us: \[ a_m = 6 + 9m \]
This formula is a powerful tool that allows you to compute any term of the sequence directly.

Sequence Type Identification

Recognizing the type of sequence is the foundation of solving related problems. There are primarily two types: arithmetic and geometric. An arithmetic sequence has a constant difference between successive terms, while a geometric sequence has a constant ratio.
To identify the sequence type, first calculate the differences between consecutive terms. If these values are all the same, you have an arithmetic sequence. If instead, the ratios of consecutive terms are identical, then it's geometric.
In the sequence 15, 24, 33, 42, 51, we noticed that each term increases by 9, confirming it's an arithmetic sequence. Identifying this early on informs the correct approach for finding the general term and analyzing further properties of the sequence.