Problem 25 - College Algebra with Modeling and Visualization

Question

Complete the following for the recursively defined sequence. (a) Find the first four terms. (b) Graph these terms. \(a_{n}=a_{n-1}+n ; a_{1}=1\)

Step-by-Step Solution

Answer

First four terms: 1, 3, 6, 10. Graph the sequence using these points.

Step 1: Understanding the Recursive Formula

The sequence is defined recursively with the formula \(a_n = a_{n-1} + n\) and the first term as \(a_1 = 1\). This means we use the previous term \(a_{n-1}\) and add \(n\) to get the next term in the sequence.

Step 2: Find the Second Term, \(a_2\)

Using the recursive formula:\[a_2 = a_1 + 2 = 1 + 2 = 3\]Thus, the second term \(a_2\) is 3.

Step 3: Find the Third Term, \(a_3\)

Continuing with the recursive formula:\[a_3 = a_2 + 3 = 3 + 3 = 6\]Therefore, the third term \(a_3\) is 6.

Step 4: Find the Fourth Term, \(a_4\)

Apply the formula again:\[a_4 = a_3 + 4 = 6 + 4 = 10\]Thus, the fourth term \(a_4\) is 10.

Step 5: Graph the First Four Terms

Now that we have the first four terms: \(a_1 = 1\), \(a_2 = 3\), \(a_3 = 6\), and \(a_4 = 10\), we can plot them on a graph. Plot each term as a point where the x-coordinate is the term number and the y-coordinate is the value of that term.

Key Concepts

Recursive Formula

A recursive formula is a rule that defines each term of a sequence using the preceding terms. In a recursively defined sequence, you start with an initial term, often denoted as \(a_1\), and then use a formula to determine each subsequent term based on the one before it. For instance, in the sequence given in the exercise, the formula is \(a_n = a_{n-1} + n\). This tells us that to find the \(n\)-th term, we add \(n\) to the preceding term \(a_{n-1}\).

In our sequence, the initial term \(a_1\) is given as 1. We then use the recursive formula to compute the subsequent terms:

• \(a_2 = a_1 + 2 = 1 + 2 = 3\)
• \(a_3 = a_2 + 3 = 3 + 3 = 6\)
• \(a_4 = a_3 + 4 = 6 + 4 = 10\)

This recursive approach is particularly useful as it lays out a step-by-step process to find as many terms as needed from a sequence.

Sequence Graphing

Graphing a sequence helps visualize the progression and patterns of its terms. Each term in the sequence can be plotted as a point on a coordinate plane. The x-axis typically represents the term number, while the y-axis represents the term's value.

For the sequence \(a_{n}\) we calculated earlier, the first four terms \(a_1 = 1\), \(a_2 = 3\), \(a_3 = 6\), and \(a_4 = 10\), can be plotted as:

• (1, 1)
• (2, 3)
• (3, 6)
• (4, 10)

By connecting these points, you can observe the linear pattern formed by the terms. This visual representation makes it easier to understand how terms build up on one another using the recursive formula.

Term Calculation

Term calculation in a recursively defined sequence involves using the recursive formula iteratively. This process allows you to calculate not only the initial terms but extends logically to find any desired term in the sequence.

Take for instance our sequence formula: \(a_n = a_{n-1} + n\). To find \(a_5\), you will first need \(a_4\), which is known to be 10, and then calculate:
\[a_5 = a_4 + 5 = 10 + 5 = 15\] Therefore, \(a_5\) is 15. The beauty of recursively defined sequences lies in this straightforward approach, enabling calculation of any term provided the previous terms are known.

This method highlights the importance of accurately establishing initial terms like \(a_1\) and reliably computing each subsequent step with the given formula.

Sequence Terms

Sequence terms are the values in the sequence, each representing a step along the path defined by the recursive formula. Understanding the relationship between consecutive terms is key to decoding the sequence correctly.

The terms we calculated:

• \(a_1 = 1\)
• \(a_2 = 3\)
• \(a_3 = 6\)
• \(a_4 = 10\)

Each term after \(a_1\) depends directly on the previous term and incorporates the sequence's pattern dictated by adding the term number to the previous term. Recognizing patterns in sequence terms can often allow you to predict further terms without recalculating from the beginning, provided the pattern maintains consistency.

As you increase the terms, understanding these relationships becomes crucial in grasping the full scope and application of recursively defined sequences.