Problem 58 - College Algebra

Question

For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence. $$ a_{1}=0, d=4 $$

Step-by-Step Solution

Answer

Plot the points (1, 0), (2, 4), (3, 8), (4, 12), (5, 16) on a graph.

Step 1: Understand the Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. The first term is denoted by \( a_1 \) and the common difference between terms is denoted by \( d \). In this problem, we are given \( a_1 = 0 \) and \( d = 4 \).

Step 2: Calculate the Next Terms

To find each term in the sequence, use the formula for the \( n \)-th term of an arithmetic sequence: \( a_n = a_1 + (n-1) \, d \). We need to calculate the first five terms.- \( a_1 = 0 \)- \( a_2 = 0 + (2-1) \cdot 4 = 4 \)- \( a_3 = 0 + (3-1) \cdot 4 = 8 \)- \( a_4 = 0 + (4-1) \cdot 4 = 12 \)- \( a_5 = 0 + (5-1) \cdot 4 = 16 \)

Step 3: List the First 5 Terms

From our calculations, we see that the first five terms of the arithmetic sequence are: \( 0, 4, 8, 12, 16 \).

Step 4: Graph the Terms

To graph these terms, plot each point corresponding to a term in the sequence on a Cartesian plane. The x-axis will represent the term number (1, 2, 3, 4, 5) and the y-axis will represent the term's value (0, 4, 8, 12, 16). Plot the points (1, 0), (2, 4), (3, 8), (4, 12), and (5, 16) on the graph and connect them sequentially to visualize the arithmetic sequence.

Key Concepts

Common Difference

In an arithmetic sequence, a key concept is the common difference. This is the consistent interval or gap between each term in the sequence. Imagine it as the step you take along the sequence from one term to the next. In our example, the common difference is given as 4. So, every time you move from one term to the next in the sequence, you add 4. This regular interval is what makes the sequence arithmetic, showcasing a constant rate of change.

The common difference is denoted by the letter \( d \). It acts like a building block in constructing an arithmetic sequence. If you think of the sequence as a ladder, each rung is spaced evenly apart by the common difference. This is crucial for predicting or calculating further terms in the sequence. Without knowing the common difference, the sequence would be incomplete or random.

Understanding and identifying the common difference is vital because it directly influences how the sequence progresses or grows. Knowing this will help you easily calculate any term in an arithmetic sequence.

n-th Term Formula

The n-th term formula is a powerful tool for figuring out any term in an arithmetic sequence without listing all of the previous terms. The formula is given by \( a_n = a_1 + (n-1) \, d \). Here, \( a_n \) represents the term you want to find, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference. This formula allows you to jump straight to any term you need, without calculating each one sequentially.

Let's take the given example to illustrate the usage. We begin with \( a_1 = 0 \) and \( d = 4 \). To find the third term, say \( a_3 \), substitute \( n = 3 \) into the formula:
\( a_3 = 0 + (3-1)\cdot 4 = 8 \).
This shows that the third term is 8, without needing to find the second term first, though it can be useful for verification.

This formula simplifies calculations and saves time. Especially when you are asked to find terms hundreds or thousands deep in a sequence, this formula is your best friend.

Graphing Sequences

Graphing sequences visually represents the terms in an arithmetic sequence, making the constant difference clear. When graphing, the term number is usually placed on the x-axis, and the actual term values are charted on the y-axis.

For our example, we identified the first five terms as 0, 4, 8, 12, and 16. On a graph, for term 1, you plot a point at (1, 0), for term 2 at (2, 4), and so on, up to point (5, 16). These points will illustrate a straight line when connected, which is a characteristic of arithmetic sequences. The straight line reflects the constant increase by the common difference, 4, between each pair of successive points.

Imagine you are drawing stair steps on the graph, each step of the stairs representing a jump or a consistent addition of text \( d \). This visual aid provides insight and makes it easier to understand how these numbers form a pattern rather than random separate points. Graphing is a very effective learning tool; it makes it easier to see the relationships between terms in a sequence.