Problem 37 - Algebra 2

Question

Write an equation for the nth term of each arithmetic sequence. \(7,16,25,34, \dots\)

Step-by-Step Solution

Answer

The nth term equation is \(a_n = 9n - 2\).

Step 1: Identify the First Term

The first term of the sequence is often denoted by \(a_1\). In this sequence, \(a_1 = 7\). This is a crucial starting point for writing the equation.

Step 2: Determine the Common Difference

In an arithmetic sequence, each term is obtained by adding a fixed number, called the common difference, to the previous term. To find the common difference \(d\), subtract the first term from the second term: \(d = 16 - 7 = 9\).

Step 3: Write the General Formula for the nth Term

The nth term of an arithmetic sequence can be expressed as \(a_n = a_1 + (n-1) \, d\). We will substitute \(a_1 = 7\) and \(d = 9\) into this general formula.

Step 4: Substitute Values into the Formula

Substitute the identified values into the formula: \(a_n = 7 + (n-1) \, 9\).

Step 5: Simplify the Equation

Simplify the equation further by distributing the common difference: \(a_n = 7 + 9n - 9\). Then, combine like terms: \(a_n = 9n - 2\). This is the equation for the nth term of the sequence.

Key Concepts

The nth term

In mathematics, especially when dealing with sequences, the term "\(n\)th term" refers to any term in a sequence which you can find by specifying its position:\(n\). For an arithmetic sequence, this concept is particularly important.
An arithmetic sequence is a string of numbers in which each term after the first is generated by adding a constant, known as the common difference, to the previous term. So when you need to find the \(n\)th term of an arithmetic sequence, what you're really doing is putting a number into a predictable spot!

• The first term is labeled \(a_1\).
• The second term can be denoted as \(a_2\), which equals \(a_1 + d\), where \(d\) is the common difference.
• Similarly, the nth term is denoted as \(a_n\).

This method allows us to "jump" straight to any term in the sequence without listing all the previous ones!

Common difference

The heart of the arithmetic sequence lies in the common difference. This is the constant amount that gets added to each term to yield the next one in the sequence. To find the common difference, you simply subtract the first term from the second term.
In our example, the terms are: 7, 16, 25, 34, etc. So, the common difference \(d\) can be calculated as follows:

• Subtract the first term from the second: \(16 - 7 = 9\)
• Double-check with another pair of terms: \(25 - 16\) is also \(9\)
• Therefore, the common difference \(d\) is \(9\).

Understanding the common difference is crucial since it shapes the progression of the sequence, affecting the spacing between consecutive terms.

General formula

The formula for the \(n\)th term in an arithmetic sequence is like a roadmap that tells you how to travel from the first term to any term in the sequence. This is expressed using the general formula:\[a_n = a_1 + (n-1) \times d\]Where:

• \(a_n\): represents the \(n\)th term
• \(a_1\): is the first term of the sequence
• \(d\): is the common difference

When substituting the known values from our sequence example—\(a_1 = 7\) and \(d = 9\)—the formula becomes:\[a_n = 7 + (n-1) \times 9\]Simplifying it further, you get:\[a_n = 9n - 2\]This tells you that any term can be acquired by calculating \(9n - 2\), highlighting the elegance of arithmetic sequences and how they can be defined by such a simple formula.