Problem 20 - Algebra and Trigonometry

Question

\(19-24\) . 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

Terms: 2, 6, 12, 20, 30, 42, 56, 72, 90, 110. Plot these on a graph.

Step 1: Understand the Formula

The given sequence is defined by the formula \(a_n = n^2 + n\). This means for each term \(a_n\), you substitute \(n\) with the term's position in the sequence, starting from \(n=1\).

Step 2: Calculate First 10 Terms

Using the formula \(a_n = n^2 + n\), calculate the first 10 terms:- \(a_1 = 1^2 + 1 = 2\)- \(a_2 = 2^2 + 2 = 6\)- \(a_3 = 3^2 + 3 = 12\)- \(a_4 = 4^2 + 4 = 20\)- \(a_5 = 5^2 + 5 = 30\)- \(a_6 = 6^2 + 6 = 42\)- \(a_7 = 7^2 + 7 = 56\)- \(a_8 = 8^2 + 8 = 72\)- \(a_9 = 9^2 + 9 = 90\)- \(a_{10} = 10^2 + 10 = 110\)

Step 3: Graph the Sequence

Using a graphing calculator, plot the points for the first 10 terms of the sequence. The x-coordinates will be from 1 to 10, and the y-coordinates will be the corresponding sequence values: (1,2), (2,6), (3,12), (4,20), (5,30), (6,42), (7,56), (8,72), (9,90), (10,110).

Key Concepts

Using a Graphing Calculator

A graphing calculator is a powerful tool used in mathematics to perform complex calculations and to visualize data graphically. When working with sequences, a graphing calculator can simplify the process considerably. You can enter the sequence formula directly into the calculator to compute values for various terms. This can save time and help avoid errors in manual calculations.

• Powerful for handling large sequences with multiple terms.
• Enables visualization of sequences via plotting.
• Efficiently calculates each term by substituting values.

By enabling quick calculations and immediate visual feedback, a graphing calculator aids in deeper understanding of sequences and their behavior.

Calculating Sequence Terms

Calculating the terms of a sequence involves substituting different values of the variable into the sequence formula. In our exercise, the sequence is given by the formula \(a_n = n^2 + n\).For each term, replace \(n\) with its position number in the sequence starting from 1. Here are steps to find terms:

• Determine position number \(n\), starting from \(n=1\).
• Substitute \(n\) into the formula \(a_n = n^2 + n\).
• Calculate the result to find each specific term \(a_n\).

For example, the first term \(a_1\) is calculated as follows: replace \(n\) by 1 in the formula, so \(a_1 = 1^2 + 1 = 2\).By following the same process, you can calculate subsequent terms like \(a_2 = 6\), \(a_3 = 12\), and so forth, up to \(a_{10} = 110\). This systematic calculation ensures all terms are accurate and reliable.

Graphing Sequences

Graphing sequences is a way to visualize their pattern and behavior across different term positions. Once you have calculated the values of a sequence, plotting them helps in understanding how the terms relate to each other. To graph a sequence, follow these simple steps:

• Identify each sequence term and its position number.
• Use the graphing calculator to plot points where the x-coordinate is the position number and the y-coordinate is the term value.
• Connect the dots to see the pattern or trend.

For example, when plotting the first 10 terms of our sequence, you generate points like (1,2), (2,6), (3,12), (4,20), up to (10,110). Observing these points on a graph allows you to visualize how the sequence terms grow, capturing both the linear progression and any exponential growth.