Problem 10

Question

The $n$ th term of an arithmetic 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 part (a). $$a_{n}=\frac{1}{2}(n-1)$$

Step-by-Step Solution

Verified

Answer

First five terms: 0, $\frac{1}{2}$, 1, $\frac{3}{2}$, 2. Common difference $d = \frac{1}{2}$. Graph by plotting these points on a coordinate plane.

1Step 1: Understand the Formula

The formula given for the nth term of the arithmetic sequence is $a_n = \frac{1}{2}(n-1)$. This formula will help us find the terms of the sequence, and the common difference.

2Step 2: Find the First Term

To find the first term, substitute $n = 1$ into the formula: $a_1 = \frac{1}{2}(1-1) = \frac{1}{2} \times 0 = 0$. So, the first term $a_1 = 0$.

3Step 3: Find the Second Term

Substitute $n = 2$ into the formula to find the second term: $a_2 = \frac{1}{2}(2-1) = \frac{1}{2} \times 1 = \frac{1}{2}$. So, the second term $a_2 = \frac{1}{2}$.

4Step 4: Find the Third Term

Substitute $n = 3$ into the formula: $a_3 = \frac{1}{2}(3-1) = \frac{1}{2} \times 2 = 1$. Hence, $a_3 = 1$.

5Step 5: Find the Fourth Term

For the fourth term, substitute $n = 4$ into the formula: $a_4 = \frac{1}{2}(4-1) = \frac{1}{2} \times 3 = \frac{3}{2}$. Thus, $a_4 = \frac{3}{2}$.

6Step 6: Find the Fifth Term

Calculate the fifth term by substituting $n = 5$: $a_5 = \frac{1}{2}(5-1) = \frac{1}{2} \times 4 = 2$. So, $a_5 = 2$.

7Step 7: Verify Sequence and Find Common Difference

The first five terms are: $0, \frac{1}{2}, 1, \frac{3}{2}, 2$. The common difference $d$ can be calculated as the difference between consecutive terms. Thus, $d = \frac{1}{2}$.

8Step 8: Graph the Sequence

Plot the terms $(1,0), (2,\frac{1}{2}), (3,1), (4,\frac{3}{2}), (5,2)$ on a graph. The x-axis represents the terms index $n$, and the y-axis represents the term value $a_n$. Connect these points to form a straight line.

Key Concepts

Common DifferenceSequence Graphingn-th Term Formula

Common Difference

In an arithmetic sequence, the common difference $d$ is the constant amount that you add to each term to get to the next term. This constant difference is what makes the sequence "arithmetic." To find $d$, simply subtract any term from the term that comes after it. In our example, the sequence starts with the terms $0, \ \frac{1}{2}, \ 1, \ \frac{3}{2}, \ \text{and} \ 2$. By subtracting each term from the subsequent term:

$\frac{1}{2} - 0 = \frac{1}{2}$
$1 - \frac{1}{2} = \frac{1}{2}$
$\frac{3}{2} - 1 = \frac{1}{2}$
$2 - \frac{3}{2} = \frac{1}{2}$

This confirms that the common difference is $d = \frac{1}{2}$.

Sequence Graphing

Graphing an arithmetic sequence can help you visualize the growth of the sequence over time. To graph, simply create a set of coordinate points using the term's index $n$ as the x-value and the term's value $a_n$ as the y-value. In our sequence

The point for $n=1$ is $(1, 0)$
The point for $n=2$ is $(2, \frac{1}{2})$
The point for $n=3$ is $(3, 1)$
The point for $n=4$ is $(4, \frac{3}{2})$
The point for $n=5$ is $(5, 2)$

Plot these points on a graph. The x-axis will show the position of the term and the y-axis will show its value. Joining these points with a straight line will represent the arithmetic sequence visually, showing a clear, linear pattern.

n-th Term Formula

The n-th term formula $a_n = \frac{1}{2}(n-1)$ allows you to find any term in the arithmetic sequence without having to list all the previous terms. In this formula: