Problem 18 - Precalculus: Mathematical for Calculus

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

First 10 terms are 2, 6, 12, 20, 30, 42, 56, 72, 90, 110. Graph these values on a calculator.

Step 1: Define the Sequence Formula

The sequence is defined by the formula \( a_n = n^2 + n \). This means that for each term in the sequence, we will substitute a value for \( n \) into this equation to get the corresponding term value.

Step 2: Calculate First Term (n=1)

Substitute \( n = 1 \) into the sequence formula: \( a_1 = 1^2 + 1 = 2 \). So, the first term is 2.

Step 3: Calculate Second Term (n=2)

Substitute \( n = 2 \) into the sequence formula: \( a_2 = 2^2 + 2 = 6 \). So, the second term is 6.

Step 4: Calculate Third Term (n=3)

Substitute \( n = 3 \) into the sequence formula: \( a_3 = 3^2 + 3 = 12 \). So, the third term is 12.

Step 5: Calculate Fourth Term (n=4)

Substitute \( n = 4 \) into the sequence formula: \( a_4 = 4^2 + 4 = 20 \). So, the fourth term is 20.

Step 6: Calculate Fifth Term (n=5)

Substitute \( n = 5 \) into the sequence formula: \( a_5 = 5^2 + 5 = 30 \). So, the fifth term is 30.

Step 7: Calculate Sixth Term (n=6)

Substitute \( n = 6 \) into the sequence formula: \( a_6 = 6^2 + 6 = 42 \). So, the sixth term is 42.

Step 8: Calculate Seventh Term (n=7)

Substitute \( n = 7 \) into the sequence formula: \( a_7 = 7^2 + 7 = 56 \). So, the seventh term is 56.

Step 9: Calculate Eighth Term (n=8)

Substitute \( n = 8 \) into the sequence formula: \( a_8 = 8^2 + 8 = 72 \). So, the eighth term is 72.

Step 10: Calculate Ninth Term (n=9)

Substitute \( n = 9 \) into the sequence formula: \( a_9 = 9^2 + 9 = 90 \). So, the ninth term is 90.

Step 11: Calculate Tenth Term (n=10)

Substitute \( n = 10 \) into the sequence formula: \( a_{10} = 10^2 + 10 = 110 \). So, the tenth term is 110.

Step 12: List the First 10 Terms

Compile the results of the previous steps to produce the first 10 terms of the sequence: 2, 6, 12, 20, 30, 42, 56, 72, 90, 110.

Step 13: Graph the Terms on a Calculator

Using a graphing calculator, plot each term against \( n \). The \( x \)-axis will represent \( n \) (from 1 to 10), and the \( y \)-axis will represent the term's value (2, 6, 12, ..., 110). This will show a parabolic pattern in the graph as it displays a quadratic sequence.

Key Concepts

Quadratic Sequences

Quadratic sequences are a special type of number sequence where the difference between consecutive terms is not constant, but rather changes linearly. This sequence is generated from quadratic equations, most commonly of the form \( a_n = an^2 + bn + c \). In the original exercise, the quadratic formula is \( a_n = n^2 + n \), which indicates that each term is calculated by squaring the term’s position \( n \) and then adding \( n \) itself. This type of sequence is characterized by a second difference that is constant; that is, if you take the first differences of your sequence terms and then the differences of those differences (second differences), it should always be the same number. Understanding this helps in predicting the behavior of terms as \( n \) increases. This parabolic nature is why their graphs form a recognizable U-shaped curve. It’s different from a linear sequence where term differences are constant straight across.

Graphing Calculator

A graphing calculator is a powerful tool that allows you to visualize mathematical equations and analyze sequences easily. To graph a quadratic sequence like \( a_n = n^2 + n \), you'd enter the formula into the calculator. This device can plot the values against their corresponding term number \( n \). First, you'd need to compute each term value for a given number \( n \). Then, plot these values on a graph where the x-axis corresponds to \( n \) and the y-axis corresponds to the sequence term. * **Setting Up**: Input the formula into the function or equation section of your calculator.* **Graphing**: Select the range of \( n \) (in this exercise, from 1 to 10) and view the graph representation. The resulting graph of this quadratic sequence will display a parabola.* **Analysis**: Utilize calculator features to analyze points, such as finding intersections, maximum, and minimum points. This helps in understanding advanced aspects of sequences, like convergence or behavior at extreme values.

Sequence Terms

Sequence terms are the individual elements or numbers in a sequence. In the context of quadratic sequences, each term is determined based on its position in the sequence, or \( n \). For the sequence given by \( a_n = n^2 + n \), terms are calculated by plugging consecutive integers starting from 1 into the formula. Each of these calculated terms forms a unique part of the sequence.* **Calculation**: To find each term, substitute the value of \( n \) into the given formula. For example, the first term \( a_1 \) is found by calculating \( 1^2 + 1 = 2 \).* **Consistency**: These terms display consistent behavior in how they increase; each term is generally larger than the previous one, stemming from the quadratic structure.* **Application**: Understanding the value and position of sequence terms allows one to plot them in a graph for visual representation, observing how changes in \( n \) affect the term values due to the nature of quadratic functions.