Problem 20 - Precalculus Mathematics 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: 2, 6, 12, 20, 30, 42, 56, 72, 90, 110. Graphed on calculator.

Step 1: Define the Sequence Formula

The given sequence formula is \( a_n = n^2 + n \). We need to find the sequence terms by substituting different values of \( n \) into this formula.

Step 2: Calculate the First 10 Terms

Substitute \( n = 1, 2, 3, \ldots, 10 \) into the formula \( a_n = n^2 + n \), and calculate each term.- 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 \( n = 10 \), \( a_{10} = 10^2 + 10 = 110 \)

Step 3: Use Graphing Calculator for Visualization

Input or verify calculated terms in a graphing calculator: \( (1, 2), (2, 6), (3, 12), (4, 20), (5, 30), (6, 42), (7, 56), (8, 72), (9, 90), (10, 110) \). Plot these points on the graph to visualize the sequence.

Key Concepts

Quadratic Sequences

Quadratic sequences are sequences where the difference between terms changes at a constant rate. This is different from arithmetic or geometric sequences, where differences or ratios are constant. Instead, in quadratic sequences, these increments themselves form an arithmetic sequence.
The general formula for a quadratic sequence can be structured as \( a_n = an^2 + bn + c \), where \( a, b, \) and \( c \) are constants, and \( n \) is the term number. For our exercise, the formula is \( a_n = n^2 + n \). Here, each term involves squaring the sequence number \( n \) and adding \( n \) itself.
If you want to identify whether a sequence is quadratic, calculate the first and second differences between terms:

• The first difference is the difference between consecutive terms.
• The second difference is the difference between consecutive first differences.

For quadratic sequences, the second difference is constant. This unique property helps confirm a quadratic progression.

Graphing Calculators

A graphing calculator is a powerful tool that helps visualize mathematical concepts like sequences and functions. By feeding a sequence formula into a graphing calculator, you can plot the terms and get a clearer picture of their progression.
For sequences like \( a_n = n^2 + n \), a graphing calculator allows you to:

• Input a list of terms to see them plotted as points in a coordinate system.
• Understand the nature of the sequence at a glance; noticing, for instance, the parabolic (u-shaped) curve typical for quadratic sequences.
• Easily calculate more terms if needed without recalculating manually, just by extending the range of \( n \).

By plotting the first 10 terms, you can verify calculations and ensure your understanding of the term progression. Most graphing calculators offer additional functionalities such as zooming and tracing the plotted points for a more detailed examination.

Term Calculation

Calculating terms of a sequence uses a formula to find specific values for given term positions. For the given sequence \( a_n = n^2 + n \), each term is a function of \( n \), representing its position in the sequence.
The process to find terms involves:

• Substituting each term position number \( n \) into the sequence formula.
• Performing arithmetic calculations as per the formula.
• Collecting results to list terms in sequence.

As demonstrated:

• For \( n = 1 \), \( a_1 = 1^2 + 1 = 2 \)
• For \( n = 2 \), \( a_2 = 2^2 + 2 = 6 \)
• Continues up to \( n = 10 \)

Each calculated term follows the pattern of increasing differences, characteristic of quadratic sequences. Mastering term calculation helps in seamlessly generating sequence terms necessary for analysis or graphing.