Problem 1 - 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). $$a_{n}=5+2(n-1)$$

Step-by-Step Solution

Answer

First five terms: 5, 7, 9, 11, 13. Common difference: 2.

Step 1: Find the First Term

To find the first term of the sequence, substitute \(n = 1\) into the formula: \(a_1 = 5 + 2(1-1)\). Simplify to get \(a_1 = 5\).

Step 2: Find the Second Term

Substitute \(n = 2\) into the formula: \(a_2 = 5 + 2(2-1)\). Simplify to get \(a_2 = 7\).

Step 3: Find the Third Term

Substitute \(n = 3\) into the formula: \(a_3 = 5 + 2(3-1)\). Simplify to get \(a_3 = 9\).

Step 4: Find the Fourth Term

Substitute \(n = 4\) into the formula: \(a_4 = 5 + 2(4-1)\). Simplify to get \(a_4 = 11\).

Step 5: Find the Fifth Term

Substitute \(n = 5\) into the formula: \(a_5 = 5 + 2(5-1)\). Simplify to get \(a_5 = 13\).

Step 6: Identify the First Five Terms

The first five terms of the sequence are \(5, 7, 9, 11, 13\).

Step 7: Determine the Common Difference

The common difference \(d\) is the difference between consecutive terms. Calculate: \(7 - 5 = 2\), \(9 - 7 = 2\), \(11 - 9 = 2\), \(13 - 11 = 2\). Therefore, \(d = 2\).

Step 8: Graph the Terms

Plot the points \((1, 5), (2, 7), (3, 9), (4, 11), (5, 13)\) on a graph with the x-axis representing the term number and the y-axis representing the term value. The graph will show a straight line, indicating an arithmetic sequence.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial component. It determines how much each term in the sequence increases or decreases from the previous one. Think of it as the consistent step size that each term takes from the one before. For our sequence, given by the formula \( a_{n} = 5 + 2(n-1) \), it is important to find this difference to truly understand the sequence's behavior.
To calculate the common difference \( d \) in this specific sequence, subtract the first term from the second term, or any consecutive terms from each other. As calculated:

• \( 7 - 5 = 2 \)
• \( 9 - 7 = 2 \)
• \( 11 - 9 = 2 \)
• \( 13 - 11 = 2 \)

This consistent difference of 2 between each term confirms that our common difference \( d = 2 \). It's a fixed number that explains how the sequence progresses.

Sequence Graphing

Graphing an arithmetic sequence helps in visualizing the linear relationship between consecutive terms. It is usually straightforward, given the constants involved in such sequences. In our sequence, the formula \( a_{n} = 5 + 2(n-1) \) guides us in plotting points on a graph. The graph is constructed as follows:

• The horizontal axis (x-axis) represents the term number, \( n \).
• The vertical axis (y-axis) captures the value of each term in the sequence, \( a_{n} \).

By plotting the points we discovered, specifically \( (1, 5), (2, 7), (3, 9), (4, 11), (5, 13) \), the sequence reveals itself as a straight line. This line verifies the arithmetic nature of the sequence. Each point represents a term, and the uniform spacing between the points equates to the common difference of 2. This clear visualization assists in understanding the sequence's linear pattern.

First Five Terms

Determining the first five terms of an arithmetic sequence is essential for grasping its initial pattern. Following the provided formula \( a_{n} = 5 + 2(n-1) \), these terms are systematically calculated by substituting sequential values of \( n \), starting from 1.
Here's how the first five terms are derived:

• For \( n = 1 \), \( a_{1} = 5 + 2(1-1) = 5 \)
• For \( n = 2 \), \( a_{2} = 5 + 2(2-1) = 7 \)
• For \( n = 3 \), \( a_{3} = 5 + 2(3-1) = 9 \)
• For \( n = 4 \), \( a_{4} = 5 + 2(4-1) = 11 \)
• For \( n = 5 \), \( a_{5} = 5 + 2(5-1) = 13 \)

Through this calculation, we confirm the first five terms of the sequence are \( 5, 7, 9, 11, 13 \). Recognizing these terms gives insight into the sequence's initial behavior and overshadows the subsequent pattern dictated by the common difference.