Problem 9 - Algebra 2 and Trigonometry

Question

In \(9-14 :\) a. Find the common difference of each arithmetic sequence. b. Write the \(n\) th term of each sequence for the given value of \(n .\) $$ 3,6,9,12, \ldots, n=8 $$

Step-by-Step Solution

Answer

The common difference is 3, and the 8th term is 24.

Step 1: Identify the Arithmetic Sequence

The given sequence is \(3, 6, 9, 12, \ldots\). We will first identify the common difference by observing the difference between consecutive terms.

Step 2: Determine the Common Difference

To find the common difference \(d\), subtract the first term from the second term: \(6 - 3 = 3\). \(d = 3\) is the difference between each term and the preceding term.

Step 3: Use the Formula for the nth Term

The formula for the \(n\)th term of an arithmetic sequence is \(a_n = a_1 + (n-1) \times d\), where \(a_1\) is the first term of the sequence and \(d\) is the common difference.

Step 4: Calculate the nth Term

Substitute \(a_1 = 3\), \(d = 3\), and \(n = 8\) into the formula: \[a_n = 3 + (8-1) \times 3 = 3 + 21 = 24.\] Thus, the 8th term of the sequence is 24.

Key Concepts

Understanding Common Difference

In an arithmetic sequence, the common difference is a key feature. It refers to the constant amount that each term in the sequence increases by. This is what makes the sequence arithmetic and not random.
To find this difference, simply subtract any term from the term that follows it. For example, in the sequence provided:

• 3, 6, 9, 12,...

you subtract the first term from the second (6 - 3 = 3). Thus, the common difference, denoted as \(d\), is 3.
Understanding the common difference helps predict future terms in the sequence and is essential for mathematical calculations involving arithmetic sequences.

Exploring the nth Term Formula

The nth term formula is a powerful tool used in arithmetic sequences to find any term without listing all previous ones.
This formula is given by:

• \(a_n = a_1 + (n-1) \times d\)

where:

• \(a_n\) is the nth term we want to find,
• \(a_1\) is the first term in the sequence,
• \(n\) is the term position in the sequence,
• \(d\) is the common difference.

For example, in the sequence "3, 6, 9, 12,..." which has a common difference \(d = 3\):

If we want to find the 8th term, we set \(a_1 = 3\), \(d = 3\), and \(n = 8\), and plug them into the formula:
\[a_n = 3 + (8-1) \times 3 = 3 + 21 = 24\]
This efficient formula allows us to find the value of the nth term with ease.

Performing Sequence Calculation

Sequence calculation involves finding specific terms in a sequence and understanding the pattern and progression of terms.
In the exercise provided, the sequence calculation started with identifying the sequence and its common difference.

• The sequence: 3, 6, 9, 12,...

has a common difference of 3. To calculate the 8th term using the nth term formula:

• First, identify the first term \(a_1 = 3\)
• Then, note the common difference \(d = 3\)
• Finally, choose the term position \(n = 8\)

By inserting these values into the nth term formula:\[a_n = 3 + (8-1) \times 3 = 3 + 21 = 24\]
This sequence calculation confirms that the 8th term is 24. Understanding sequence calculations helps in predicting terms, identifying patterns, and solving real-life problems linked to arithmetic sequences.