Problem 2 - Algebra and Trigonometry

Question

1–4 A sequence is given. (a) Find the first five terms of the sequence. (b) What is the common difference \(d ?\) (c) Graph the terms you found in (a). $$a_{n}=3-4(n-1)$$

Step-by-Step Solution

Answer

The first five terms are 3, -1, -5, -9, and -13; the common difference is -4.

Step 1: Understand the Formula

The sequence is given by the formula \( a_n = 3 - 4(n-1) \). This formula will help us find the terms in the sequence by substituting the values of \( n \).

Step 2: Find the First Term

Substitute \( n = 1 \) into the formula: \( a_1 = 3 - 4(1-1) = 3 - 0 = 3 \). So the first term is 3.

Step 3: Find the Second Term

Substitute \( n = 2 \) into the formula: \( a_2 = 3 - 4(2-1) = 3 - 4 = -1 \). The second term is -1.

Step 4: Find the Third Term

Substitute \( n = 3 \) into the formula: \( a_3 = 3 - 4(3-1) = 3 - 8 = -5 \). The third term is -5.

Step 5: Find the Fourth Term

Substitute \( n = 4 \) into the formula: \( a_4 = 3 - 4(4-1) = 3 - 12 = -9 \). The fourth term is -9.

Step 6: Find the Fifth Term

Substitute \( n = 5 \) into the formula: \( a_5 = 3 - 4(5-1) = 3 - 16 = -13 \). The fifth term is -13.

Step 7: Identify the Sequence

The first five terms of the sequence are 3, -1, -5, -9, and -13.

Step 8: Find the Common Difference

The common difference \( d \) is the difference between successive terms. Calculate: \( d = a_2 - a_1 = -1 - 3 = -4 \). The common difference is \( -4 \).

Step 9: Graph the Terms

On a graph, plot the points: (1, 3), (2, -1), (3, -5), (4, -9), (5, -13). These points form a line, indicating an arithmetic sequence with a common difference of \( -4 \).

Key Concepts

Common Difference

In an arithmetic sequence, one of the most important concepts is the **common difference**. This is the constant number added to each term to get the next term in the sequence. It plays a crucial role in understanding how the sequence progresses.

To find the common difference, you subtract any term in the sequence from the term that follows it. For example, in the sequence given by:

• 3, -1, -5, -9, -13

We can find the common difference by taking the second term, -1, and subtracting the first term, 3:

• \[d = a_2 - a_1 = -1 - 3 = -4\]

This result, -4, is the common difference. Each subsequent term is obtained by adding -4 to the previous term. Understanding this pattern is key to working with and predicting arithmetic sequences.

Sequence Graphing

Graphing an arithmetic sequence can offer a visual representation that makes it easier to understand the sequence's behavior. When you graph the terms of an arithmetic sequence, each point on the graph corresponds to a term in the sequence.

Take, for instance, the sequence with terms: 3, -1, -5, -9, -13, which arise from the formula \(a_n = 3 - 4(n-1)\).
Begin by plotting each term:

• (1, 3)
• (2, -1)
• (3, -5)
• (4, -9)
• (5, -13)

These points will create a straight line when connected. This linear pattern is a hallmark of an arithmetic sequence, reflecting a constant change in terms due to the common difference. The slope of the line corresponds to the common difference, which in this sequence is -4. The downward slope highlights the decreasing nature of the sequence.

Term Calculation

Calculating terms in an arithmetic sequence relies on applying a formula that accounts for both the initial term and the common difference. In our example, the given formula is \(a_n = 3 - 4(n-1)\). This is an explicit formula useful for finding any term in the sequence.

Here's how it works:

• The formula starts with 3, representing the first term.
• The expression \(-4(n-1)\) adjusts each subsequent term by decreasing it further, utilizing the common difference \(-4\).

For instance, to find the third term, set \(n\) to 3 in the formula:

• \[a_3 = 3 - 4(3-1) = 3 - 8 = -5\]

This calculation illuminates how each term is derived. The step-by-step substitution helps ensure you always find the correct result, regardless of which term you need to calculate. Being able to accurately use this formula is essential for mastering arithmetic sequences.