Problem 11 - 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 \(n .\) $$ 18,16,14,12, \ldots, n=10 $$

Step-by-Step Solution

Answer

The common difference is -2, and the 10th term is 0.

Step 1: Identifying the Common Difference

To find the common difference of an arithmetic sequence, subtract any term from the previous term. Starting with the sequence given: 18, 16, 14, 12, ..., you can see the calculations as follows:\[16 - 18 = -2 \]So, the common difference \(d\) is \(-2\).

Step 2: Understanding the Arithmetic Sequence Formula

The formula for the \(n\) th term \(a_n\) of an arithmetic sequence is:\[ a_n = a_1 + (n-1) imes d \]where \(a_1\) is the first term of the sequence and \(d\) is the common difference. For our sequence, \(a_1 = 18\) and \(d = -2\).

Step 3: Substituting Values to Find the 10th Term

Using the nth-term formula \(a_n = a_1 + (n-1) imes d\), substitute \(a_1 = 18\), \(d = -2\), and \(n = 10\):\[a_{10} = 18 + (10-1) imes (-2)\]

Step 4: Simplifying the Expression

Now, calculate the value:\[a_{10} = 18 + 9 imes (-2) \]\[a_{10} = 18 - 18 \]\[a_{10} = 0 \]

Step 5: Summarizing the Results

The common difference for the sequence is \(-2\), and the 10th term \(a_{10}\) is 0.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key feature. It helps determine how each term in the sequence is generated from the previous one. Simply put, it is the amount you add or subtract to go from one term to the next. In our example, the sequence given is 18, 16, 14, 12, ..., and you can find the common difference by:

• Subtraction: Choose any term and subtract it from the term before it.
• Example: 16 - 18 = -2

The common difference, denoted as \(d\), is \(-2\). A negative value means the sequence decreases as it progresses. If it were positive, the sequence would increase. Knowing the common difference is the first step in understanding the pattern of an arithmetic sequence.

Nth Term Formula

The nth term formula is an essential tool for finding any term in an arithmetic sequence without listing all terms one by one. This formula helps efficiently calculate the sequence's terms by relating them to their position number, \(n\). The formula for the nth term \(a_n\) is:\[ a_n = a_1 + (n-1) \times d \]

• \(a_1\) is the first term in the sequence.
• \(d\) represents the common difference.

To use the formula, simply plug in the values of the first term, the common difference, and the term number \(n\) you want to find.In our sequence, \(a_1\) is 18 and \(d\) is -2. Once you know these, calculating any term becomes straightforward with the nth term formula.

Sequence Calculation

Sequence calculation involves using the nth term formula and plugging in the correct values to find the desired term in the sequence. Following the example provided with \(n = 10\), we use the known formula for the nth term, which is \(a_n = a_1 + (n-1) \times d\). Here’s how to calculate the 10th term:

• First term \(a_1 = 18\).
• Common difference \(d = -2\).
• Substitute \(n = 10\) into the formula.\[ a_{10} = 18 + (10-1) \times (-2) \]
• Now calculate:
  • \( a_{10} = 18 + 9 \times (-2) \)
  • \( a_{10} = 18 - 18 \)
  • \( a_{10} = 0 \)

The 10th term in the sequence is 0. This accurate and quick method highlights the power of the nth term formula in arithmetic sequence calculations.

Algebra 2

In Algebra 2, sequences such as arithmetic sequences are extensively explored to build a foundation for more complex mathematical concepts. This level of algebra takes the basic idea of sequences, like our arithmetic sequence example, and links them to functions, series, and real-world problem-solving. - Algebra 2 introduces tools like the nth term formula that simplify working with sequences. - Real-world applications can include financial calculations such as calculating interest, where sequences help predict future values. Practicing these concepts prepares students for advanced studies by honing their problem-solving and analytical skills. Sequential thinking is a stepping stone towards the deeper algebraic understanding needed for calculus and higher mathematics.