Problem 42 - Precalculus with Limits

Question

In Exercises 37-42, use a graphing utility to graph the first 10 terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = \dfrac{3n^2}{n^2 + 1} \)

Step-by-Step Solution

Answer

After calculating the first 10 terms of the sequence, plot these pairs of values (1, Term1), (2, Term2), ..., (10, Term10) on a 2D graph, with \( n \) values along the \( x \)-axis and sequence terms along the \( y \)-axis.

Step 1: Find the first 10 terms of the sequence

For \( n = 1, 2, 3, \ldots, 10 \), substitute these values into the formula \( a_n = \dfrac{3n^2}{n^2 + 1} \) to get the first ten terms of this sequence.

Step 2: Pair each term with its corresponding number

After calculating the first ten terms of the sequence, assign \( n = 1 \) to the first term, \( n = 2 \) to the second term, and so on, to get pairs of values to be plotted on the graph.

Step 3: Plot the points on the graph

Using any graphing tool available, the points corresponding to the sequence terms and \( n \) values should be plotted on a simple two-dimensional graph. The \( x \)-axis will represent the index \( n \), and the \( y \)-axis will represent the value of the term of the sequence at that index.

Key Concepts

Sequence Terms

When exploring sequences in math, understanding sequence terms is essential. A sequence is an ordered list of numbers where each number is called a term. In this exercise, the sequence is given by the formula \( a_n = \frac{3n^2}{n^2 + 1} \). This formula allows us to compute each term by substituting different values of \( n \).

• For every positive integer \( n \), substitute \( n \) into the equation to find \( a_n \).
• For example, when \( n = 1 \), \( a_1 \) is calculated as \( \frac{3(1)^2}{(1)^2 + 1} = \frac{3}{2} \).
• Similarly, calculate the next terms, continuing with \( n = 2, 3, \ldots, 10 \).

The formula effectively maps \( n \) to \( a_n \), and this process of substitution is how we discover each term in the sequence.

Graphing Utilities

Graphing utilities are essential tools that help visualize mathematical concepts, such as sequences, more intuitively. By representing sequence terms graphically, one can analyze patterns and trends that may not be immediately obvious in a list format.

• To graph the sequence \( a_n = \frac{3n^2}{n^2 + 1} \), assign each \( n \) a corresponding \( a_n \) value.
• A graphing tool plots these pairs \((n, a_n)\) on a coordinate plane, where \( n \) is the x-coordinate, and \( a_n \) is the y-coordinate.
• This visual representation helps in identifying whether the sequence is increasing, decreasing, oscillating, or stabilizing at certain levels.

Using graphing utilities can significantly enhance understanding, especially when exploring complex sequences.

Precalculus

Precalculus lays the foundation for understanding more advanced mathematics, including calculus. It encompasses a variety of topics, including sequences and how to represent them graphically. In this precalculus problem, you're required to utilize both your knowledge of sequences and graphing.

• Precalculus often involves exploring how sequences progress, as sequences are an integral part of understanding limits and functions.
• Students learn to analyze how functions and sequences are behaviorally similar, which prepares them for calculus concepts like derivatives and integrals.
• This exercise combines practical application with theory, a hallmark of precalculus that strengthens the student's problem-solving toolbox.

Understanding sequences in precalculus is beneficial as it introduces concepts that will be pivotal in later studies of calculus.