Problem 42 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{1}=-3, d=5$$

Step-by-Step Solution

Answer

The general term is \(a_n = 5n - 8\).

Step 1: Understanding the Sequence

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant, called the common difference. In this problem, the first term of the sequence is given as \(a_1 = -3\) and the common difference \(d = 5\).

Step 2: Writing the General Formula for Arithmetic Sequence

The general term of an arithmetic sequence can be found using the formula:\[ a_n = a_1 + (n-1) imes d \]where \(a_n\) is the nth term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

Step 3: Substitute the Known Values

Substitute the given values into the general formula:\[ a_n = -3 + (n-1) imes 5 \]

Step 4: Simplify the Expression

Now simplify the expression:\[a_n = -3 + 5(n-1) \]Distribute the 5:\[a_n = -3 + 5n - 5 \]Combine the constant terms:\[a_n = 5n - 8 \]

Step 5: Final General Term

Thus, the general term for the arithmetic sequence is \(a_n = 5n - 8\). This expression allows you to find the nth term of the sequence by substituting the desired term number into \(n\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the consistent interval that is added to each term to get the next one. It's what makes arithmetic sequences predictable. In our exercise, the common difference, denoted by the letter \(d\), is given as 5.
This means that if you start with the first term, \(a_1 = -3\), you will keep adding 5 to each previous number to get the next number in the series.

For example:

• The second term, \(a_2\), would be \(-3 + 5 = 2\).
• The third term, \(a_3\), would be \(2 + 5 = 7\).
• This pattern continues.

Understanding the common difference is key to constructing and analyzing arithmetic sequences.

General Term Formula

The general term formula of an arithmetic sequence lets you find any term in the sequence without listing all the terms before it. This formula is:\[ a_n = a_1 + (n-1) \times d \]where:

• \(a_n\) is the nth term you want to find,
• \(a_1\) is the first term of the sequence,
• \(d\) is the common difference,
• \(n\) is the term’s position in the sequence.

In our example, substituting the known values:\(-3\) for \(a_1\) and \(5\) for \(d\):
\[ a_n = -3 + (n-1) \times 5 \]
This formula is fantastic because it saves time and energy by allowing you to directly calculate any term in the sequence using simple arithmetic steps.

Sequence Pattern

The pattern in an arithmetic sequence is driven by the constant addition of the common difference, \(d\). This creates a linear pattern where each term is spaced equally apart, making it easy to predict the sequence as it progresses.
To identify the pattern, start from the first term, and add the common difference repeatedly. This cumulative addition gives rise to a sequence of numbers with a predictable arrangement.

• First term: \(-3\)
• Second term: \(-3 + 5 = 2\)
• Third term: \(2 + 5 = 7\)
• Fourth term: \(7 + 5 = 12\)

The sequence pattern is thus \(-3, 2, 7, 12, \ldots\). Each term is simply the result of continuing this addition process. The beauty of the sequence pattern is in its simple yet powerful regularity, which is defined by the common difference.