Problem 4 - 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). $$a_{n}=\frac{1}{2}(n-1)$$

Step-by-Step Solution

Answer

The first five terms are 0, \(\frac{1}{2}\), 1, \(\frac{3}{2}\), and 2; the common difference \(d\) is \(\frac{1}{2}\).

Step 1: Identify the Formula

The sequence is given by the formula \(a_{n}=\frac{1}{2}(n-1)\). We will use this formula to find the first five terms.

Step 2: Calculate the First Term

Substitute \(n=1\) into \(a_{n}=\frac{1}{2}(n-1)\) to find the first term: \(a_{1}=\frac{1}{2}(1-1)=0\).

Step 3: Calculate the Second Term

Substitute \(n=2\) into \(a_{n}=\frac{1}{2}(n-1)\) to find the second term: \(a_{2}=\frac{1}{2}(2-1)=\frac{1}{2}\).

Step 4: Calculate the Third Term

Substitute \(n=3\) into \(a_{n}=\frac{1}{2}(n-1)\) to find the third term: \(a_{3}=\frac{1}{2}(3-1)=1\).

Step 5: Calculate the Fourth Term

Substitute \(n=4\) into \(a_{n}=\frac{1}{2}(n-1)\) to find the fourth term: \(a_{4}=\frac{1}{2}(4-1)=\frac{3}{2}\).

Step 6: Calculate the Fifth Term

Substitute \(n=5\) into \(a_{n}=\frac{1}{2}(n-1)\) to find the fifth term: \(a_{5}=\frac{1}{2}(5-1)=2\).

Step 7: List the First Five Terms

The first five terms of the sequence are: 0, \(\frac{1}{2}\), 1, \(\frac{3}{2}\), and 2.

Step 8: Determine the Common Difference

The common difference \(d\) is found by subtracting consecutive terms. Here, \(d = a_{2} - a_{1} = \frac{1}{2} - 0 = \frac{1}{2}\).

Step 9: Graph the Terms

To graph the first five terms, plot the points: (1,0), (2,\(\frac{1}{2}\)), (3,1), (4,\(\frac{3}{2}\)), and (5,2) on a coordinate plane, where the x-axis represents \(n\) and the y-axis represents \(a_n\).

Key Concepts

Common Difference

Understanding the concept of the common difference is crucial in arithmetic sequences. In an arithmetic sequence, the common difference is a constant amount that separates each term from the next term in the sequence. This difference, often represented by the letter \(d\), stays the same as you move from one term to the next. In practical terms, you calculate the common difference by subtracting a term from its succeeding term.

In the example given with the formula \(a_{n} = \frac{1}{2}(n-1)\), once we calculate the terms, we notice the pattern: 0, \(\frac{1}{2}\), 1, \(\frac{3}{2}\), and 2. Here, the difference between consecutive terms is always \(\frac{1}{2}\). Thus, the common difference \(d = \frac{1}{2}\) means that each term increases by \(\frac{1}{2}\) from the previous one.

Identifying the common difference helps predict future terms in the sequence. For instance, after calculating a few terms and the common difference, finding further terms becomes straightforward by simply adding \(d\) to the latest known term.

Sequence Terms

Sequence terms are the individual values that make up a sequence. In arithmetic sequences, these terms are generated by a specific formula. The given formula in the task, \(a_{n} = \frac{1}{2}(n-1)\), is used to determine each term by substituting \(n\) with integers starting from 1.

• First term: Set \(n=1\), calculate \(a_{1} = \frac{1}{2}(1-1)=0\).
• Second term: Set \(n=2\), calculate \(a_{2} = \frac{1}{2}(2-1)=\frac{1}{2}\).
• Third term: Set \(n=3\), calculate \(a_{3} = \frac{1}{2}(3-1)=1\).
• Fourth term: Set \(n=4\), calculate \(a_{4} = \frac{1}{2}(4-1)=\frac{3}{2}\).
• Fifth term: Set \(n=5\), calculate \(a_{5} = \frac{1}{2}(5-1)=2\).

These terms form a sequence because they follow a specific logic defined by the formula. Understanding how to compute these values is fundamental to grasping more complex arithmetic topics such as constructing or deconstructing sequences.

Graphing Sequences

Graphing sequences provides a visual understanding of the behavior of sequences and their terms. By plotting the terms on a graph, you can observe patterns and verify concepts such as the common difference. In our case, we focus on plotting points that represent the sequence terms on a coordinate plane.

Each term from the sequence corresponds to a point on the plane with coordinates \((n, a_{n})\). For example:

• The first term \(a_{1} = 0\) appears as point (1,0)
• The second term \(a_{2} = \frac{1}{2}\) appears as point (2,\(\frac{1}{2}\))
• The third term \(a_{3} = 1\) appears as point (3,1)
• The fourth term \(a_{4} = \frac{3}{2}\) appears as point (4,\(\frac{3}{2}\))
• The fifth term \(a_{5} = 2\) appears as point (5,2)

When these points are plotted and connected, they reveal a straight line indicating the sequence is arithmetic. This is possible due to the uniform increase from one term to the next, reflected by the constant common difference and indicating linear growth.