Question

Describe the pattern, write the next term, and write a rule for the \(\boldsymbol{n}\) th term of the sequence. \(9,16.8,24.6,32.4, \ldots\)

Short answer

The pattern is an addition of 7.8 to each term. The next term would be 40.2. A rule for the nth term of the sequence is \(a_n = 9 + (n-1) * 7.8\).

Step-by-step solution

Determine the Common Difference

By subtracting the first term from the second, then the second term from the third, we can figure out whether the sequence has a common difference. In this case, \(16.8 - 9 = 7.8\), and \(24.6 - 16.8 = 7.8\), confirming a common difference of 7.8.

Predict the Next Term

Using the identified common difference, the next term can be predicted by adding 7.8 to the last given term, \(32.4 + 7.8 = 40.2\)

Formulate a Rule for the nth Term

Arithmetic sequences can be represented with the formula \(a_n = a_1 + (n-1) * d\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. Substituting our sequence's values we get: \(a_n = 9 + (n-1) * 7.8\)

Key concepts

Common Difference

When we talk about arithmetic sequences, the term 'common difference' refers to the constant amount that each term in the sequence is increased or decreased by, to get from one term to the next. This difference is the key to understanding how arithmetic sequences work.

For instance, consider the sequence given: 9, 16.8, 24.6, 32.4, ... Here, if we subtract the first term from the second, we find that 16.8 minus 9 equals 7.8. Repeating this subtraction between subsequent terms, such as 24.6 minus 16.8, we consistently get the same result of 7.8. This confirms that our sequence has a common difference of 7.8. It's crucial for students to accurately determine the common difference, as it is used for predicting subsequent terms and understanding the overall pattern of the sequence.

Nth Term Formula

The power of arithmetic sequences lies in the predictability of their patterns. Once we know the common difference, we can derive a general formula to calculate any term in the sequence, known as the 'nth term formula'. For a sequence starting with the first term, this formula is usually written as:
\[a_n = a_1 + (n-1) \times d\]
Here, \(a_n\) represents any term we wish to find, also referred to as the 'nth term'. The \(a_1\) is the first term in the sequence, \(n\) stands for the position of the term within the sequence, and \(d\) represents the common difference. In our exercise, with the sequence starting at 9 and a common difference of 7.8, the formula to find any term will be:\[a_n = 9 + (n-1) \times 7.8\]
This formula is essential for students, particularly when tackling problems that involve large sequences where manual counting is impractical.

Sequence Patterns

Understanding sequence patterns is fundamental to grasping the concept of arithmetic sequences. These patterns are orderly lists of numbers that are created based on a specific rule that applies uniformly across the entire sequence.

In the context of arithmetic sequences, this rule is the addition or subtraction of the common difference. Observing the starting number (known as the first term) and how it is transformed by the common difference provides insight into the sequence's behavior. For example, in our sequence (9, 16.8, 24.6, 32.4, ...), each term after the first is formed by adding 7.8 to the previous term, which reveals the pattern.

Recognizing and characterizing these patterns are not only crucial for continuing the sequence but also for understanding functions and predicting behaviors in maths and related disciplines. Being able to discern patterns assists students in identifying the structure within sequences, greatly simplifying their complexity and making it easier to work with them in various mathematical contexts.