Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. First term is 5, common difference is \(6,\) find the \(8^{\text {th }}\) term.

Step-by-Step Solution

Answer

The 8th term is 47.

Step 1: Understanding the Formula

In an arithmetic sequence, the formula to find the n-th term is given by: \( a_n = a + (n-1) imes d \) where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number you want to find.

Step 2: Substitute the Values

We are given that the first term \( a = 5 \), the common difference \( d = 6 \), and we want to find the \( 8^{\text{th}} \) term. Substitute these values into the formula: \( a_8 = 5 + (8-1) imes 6 \).

Step 3: Simplify and Compute

First, calculate \( (8-1) imes 6 \): \( 7 imes 6 = 42 \). Now, add this result to the first term: \( a_8 = 5 + 42 = 47 \). So, the 8th term is 47.

Key Concepts

Common Difference

In an Arithmetic Sequence, the common difference, often denoted by \( d \), is a pivotal concept. This is the constant value that you add to each term to get to the next term in the sequence. Understanding the common difference is crucial because it defines the relationship between the terms in the sequence.

• To find the common difference, subtract any term from the term that follows it.
• For example, if you have terms like 3, 7, 11, and 15, the common difference \( d \) is 4 because \( 7 - 3 = 4 \).
• It stays the same throughout the sequence, ensuring regularity and predictability.

In exercises, identifying the common difference helps you pin down any term in the sequence easily.

N-th Term Formula

Knowing how to find any term in an arithmetic sequence is made easy with the N-th Term Formula. This formula allows you to skip directly to the term position you need without listing all preceding terms.
The formula is expressed as:\[ a_n = a + (n-1) imes d \]

• Here, \( a \) is the first term of the sequence.
• \( n \) is the position of the term you are seeking.
• \( d \) is the common difference.

To use the formula, plug in the values you know, and solve for \( a_n \). This formula is powerful because it lets you compute any term in the sequence directly, without tedious calculations of all previous terms.

Arithmetic Sequence Formula

The Arithmetic Sequence Formula is the backbone of solving problems related to sequences. It guides you in determining any term and understanding the sequence as a whole.
The generic formula for calculating the n-th term, \( a_n \), is:\[ a_n = a + (n-1) imes d \]This captures the essence of arithmetic sequences – their linear progression and the ease of finding individual terms.

• Start by knowing the first term, denoted as \( a \).
• Use the common difference, \( d \), to determine the gap between sequential terms.
• Put these values into the formula to find any term \( a_n \).

This formula is helpful because it handles any term position with precision and ease, allowing for quick and reliable calculations in mathematical exercises.