Problem 25 - Algebra for College Students

Question

Find the required term for each arith- metic sequence. The 15 th term of \(3,8,13,18, \ldots\) 73

Step-by-Step Solution

Answer

The 15th term is 73.

Step 1: Identify Sequence Components

To find the required term of an arithmetic sequence, first identify the common difference and the first term. Here, the first term \(a_1\) is 3, and the common difference \(d\) is \(8 - 3 = 5\).

Step 2: Apply Formula for the n-th Term

The formula to find the n-th term, \(a_n\), of an arithmetic sequence is: \( a_n = a_1 + (n-1)d \).

Step 3: Substitute Values

Substitute the values into the formula: \( n = 15 \), \( a_1 = 3 \), and \( d = 5 \). This gives: \( a_{15} = 3 + (15-1) \times 5 \).

Step 4: Calculate

Simplify and calculate the expression: \( a_{15} = 3 + 14 \times 5 = 3 + 70 = 73 \).

Key Concepts

nth term formula

The nth term formula is a powerful tool used to find any term in an arithmetic sequence. In such sequences, there is always a common difference between consecutive terms. This formula helps us skip counting each term one by one and allows us to jump directly to any desired term. The formula is expressed as:\[ a_n = a_1 + (n-1)d \]Where:

• \(a_n\) represents the nth term of the sequence.
• \(a_1\) is the first term of the sequence.
• \(n\) is the position of the term you want to find within the sequence.
• \(d\) is the common difference, which remains constant throughout the sequence.

By substituting the known values into this formula, you can easily calculate any specific term in the sequence without having to list all the preceding terms. This efficiency makes it an indispensable equation in the study of arithmetic sequences.

common difference

The common difference is a crucial component of any arithmetic sequence. It is defined as the difference between any two successive terms in the sequence. To find it, simply subtract any term from the term following it. For instance, in the sequence \(3, 8, 13, 18, \ldots\), the common difference \(d\) is calculated as \(8 - 3 = 5\).This value is vital because it determines the pattern and behavior of the sequence:

• If the common difference is positive, the sequence increases with each term.
• If it is negative, each term decreases, moving in a descending fashion.
• A zero difference implies a constant sequence where all terms are equal.

Recognizing the common difference will help you understand how the sequence grows or shrinks, thus aiding in the application of the nth term formula.

sequence components

An arithmetic sequence is characterized by specific components that define its structure. These components include the first term, the common difference, and a general rule. Let's break these down:- **First term (\(a_1\))**: This is the starting point of the sequence. It is the term from which all other terms originate. In our example, the first term is 3.- **Common difference (\(d\))**: As discussed, this is the fixed amount added or subtracted as you move from one term to the next. In the example sequence, \(d = 5\).When you understand these components, solving for any term in the sequence becomes straightforward. With knowledge of the first term and the common difference, you can predict the sequence's future values using the nth term formula. This relationship between the components allows you to fully grasp and manipulate the sequence, making arithmetic sequences both predictable and easy to work with.