Problem 18 - Algebra and Trigonometry

Question

Use a graphing calculator to do the following. (a) Find the first 10 terms of the sequence. (b) Graph the first 10 terms of the sequence. \(a_{n}=n^{2}+n\)

Step-by-Step Solution

Answer

The first 10 terms are 2, 6, 12, 20, 30, 42, 56, 72, 90, and 110. These terms form a quadratic curve when graphed.

Step 1: Understand the Sequence

The sequence you're working with is defined by the formula \[ a_n = n^2 + n \]This means every term in the sequence is calculated by plugging the term number \( n \) into the formula.

Step 2: Calculate the First 10 Terms

To find the first 10 terms of the sequence, plug in the values from \( n = 1 \) to \( n = 10 \) into the formula:- For \( n = 1 \), \( a_1 = 1^2 + 1 = 2 \).- For \( n = 2 \), \( a_2 = 2^2 + 2 = 6 \).- For \( n = 3 \), \( a_3 = 3^2 + 3 = 12 \).- For \( n = 4 \), \( a_4 = 4^2 + 4 = 20 \).- For \( n = 5 \), \( a_5 = 5^2 + 5 = 30 \).- For \( n = 6 \), \( a_6 = 6^2 + 6 = 42 \).- For \( n = 7 \), \( a_7 = 7^2 + 7 = 56 \).- For \( n = 8 \), \( a_8 = 8^2 + 8 = 72 \).- For \( n = 9 \), \( a_9 = 9^2 + 9 = 90 \).- For \( n = 10 \), \( a_{10} = 10^2 + 10 = 110 \).Thus, the first 10 terms are: \( 2, 6, 12, 20, 30, 42, 56, 72, 90, 110 \).

Step 3: Graph the First 10 Terms Using a Calculator

Use your graphing calculator to plot the points corresponding to the first 10 terms of the sequence. Each point is given by \( (n, a_n) \).- Enter the sequence formula \( y = n^2 + n \) into the calculator.- Set \( n \) values from 1 to 10.- Check the consecutive points: \( (1, 2), (2, 6), (3, 12), (4, 20), (5, 30), (6, 42), (7, 56), (8, 72), (9, 90), (10, 110) \).- Observe the shape of the graph, which should be a quadratic curve.

Key Concepts

Quadratic Functions

A quadratic function is an important mathematical concept represented by the equation \[ f(x) = ax^2 + bx + c \]where \( a, b, \) and \( c \) are constants. Quadratic functions form a parabola when graphed, which is generally either "U" shaped or "n" shaped.
In our exercise, the sequence formula \( a_n = n^2 + n \) is a specific type of quadratic function where:

• \( a = 1 \), meaning the parabola opens upwards.
• \( b = 1 \), contributing to the linear term.
• \( c = 0 \), there is no constant term in our sequence.

This quadratic nature means that as \( n \) increases, the values of \( a_n \) grow larger at an increasing rate. This is evident in the step-by-step solution where terms increase from 2 to 110.
Understanding this helps when predicting the behavior of the sequence or further analyzing the pattern of the terms.

Graphing Calculator

A graphing calculator is a powerful tool that allows you to plot and visualize equations like quadratic functions easily. When using a graphing calculator, you plot the equation by:

• Entering the sequence or function formula into the calculator.
• Setting the values over which you wish to evaluate, in this case, \( n = 1 \) to \( n = 10 \).
• Viewing the plotted points and graph shape on the display screen.

For the sequence \( a_n = n^2 + n \), the graph shows individual points that align to form a parabolic curve.
This visualization helps in understanding the growth pattern of the terms. Each plotted point \((n, a_n)\) represents a term from the sequence. Observing these plotted points as a whole gives insights into the overall structure and behavior of the sequence, such as its consistent acceleration or progression as seen in the rising curve.
Graphing helps connect the algebraic formulation of sequences with their geometric representation, making complex concepts more intuitive.

Term Calculation

Term calculation involves finding specific terms of a sequence using a given formula. In the exercise, you used the quadratic equation\[ a_n = n^2 + n \]
Here’s how to systematically find the terms:

• Identify your term number \( n \).
• Substitute \( n \) into the formula.
• Calculate using the operations of squaring \( n \) and adding \( n \).

Practically, for \( n = 1 \), \( a_1 = 1^2 + 1 = 2 \); continue this method up to \( n = 10 \).
This step-by-step substitution not only helps in finding each term correctly but also highlights how the formula dictates the value of each term.
Repetition of this process builds a clearer understanding of how sequences operate and shows how even a simple change in a formula can lead to vastly different sequences in mathematics. Using this calculated information, you can fill sequences with numerous terms, assess patterns, and develop predictions.