Problem 15 - A Graphical Approach to College Algebra

Question

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{3}=2, d=1$$

Step-by-Step Solution

Answer

\(a_8 = 7\), \(a_n = n-1\).

Step 1: Understanding the formula for arithmetic sequences

The general formula for the n-th term of an arithmetic sequence is given by \( a_n = a_1 + (n-1) imes d \), where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference between successive terms.

Step 2: Finding the first term \(a_1\) using \(a_3\)

We know \( a_3 = 2 \) and \( d = 1 \). To find \( a_1 \), use the formula for the third term: \( a_3 = a_1 + (3-1) \times d = a_1 + 2 2 = a_1 + 2 \Rightarrow a_1 = 0 \).

Step 3: Calculating \(a_8\)

With \( a_1 = 0 \) and \( d = 1 \), use the formula for \( a_8 \): \( a_8 = a_1 + (8-1) \times d = 0 + 7 \times 1 = 7 \).

Step 4: Writing the expression for \(a_n\)

Using \( a_1 = 0 \) and the difference \( d = 1 \), the formula for the n-th term is \( a_n = 0 + (n-1) \times 1 = n-1 \).

Key Concepts

Understanding the n-th Term Formula

Arithmetic sequences are a type of mathematical sequence where each term increases by the same amount, called the common difference. The n-th term of an arithmetic sequence is determined using a specific formula:

• \( a_n = a_1 + (n-1) \times d \)

In this formula, \( a_n \) stands for the n-th term of the sequence, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) represents the common difference, the constant amount added to each term to get to the next term.
This formula is particularly useful because it allows us to calculate any term in the sequence without having to calculate all previous terms. For example, if the first term of a sequence is \( a_1 \), and the common difference is \( d \), we can directly find the eighth term \( a_8 \) using the formula without needing the individual values of terms 2 through 7.

What is the Common Difference?

The common difference, symbolized by \( d \), is a vital part of understanding arithmetic sequences. It tells us how much we add to each term to get the next one. This consistent addition of the same value is what defines an arithmetic sequence. In mathematics, it's crucial to distinguish between arithmetic sequences and other types such as geometric sequences, where terms are multiplied rather than added.
For instance, if you start with a number like 0 and add 1 consistently, your arithmetic sequence looks like 0, 1, 2, 3, and so on. Here, the common difference \( d \) is 1. This concept of constant change makes it easier to predict future terms. Knowing \( d \) helps in fast calculations, where you can hop from the beginning to any term in the sequence just by counting \( d \) steps.

Calculating Specific Terms in a Sequence

To find a specific term in an arithmetic sequence, we make use of the n-th term formula:

• \( a_n = a_1 + (n-1) \times d \)

Let's say you want to find the eighth term, \( a_8 \), of a sequence where the common difference \( d \) is 1 and the first term \( a_1 \) is 0. Plugging these values into the formula:

• \( a_8 = 0 + (8-1) \times 1 \)

This simplifies to \( a_8 = 0 + 7 \), which equals 7.
Once you know \( a_1 \) and \( d \), calculating any n-th term becomes straightforward. You adjust the formula based on the term you wish to find, making arithmetic sequences easy to work with. With practice, this process becomes almost automatic!