Problem 16 - College Algebra

Question

In Exercises \(15-22,\) find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\) Find \(a_{16}\) when \(a_{1}=9, d=2\)

Step-by-Step Solution

Answer

The 16th term of the arithmetic sequence is 39.

Step 1: Substitute the values into the formula

Put the given values into the formula \(a_{n} = a_{1} + (n-1)d\). Here, \(a_{1} = 9\), \(d = 2\) and \(n = 16\). So we get \(a_{16} = 9 + (16-1)2\).

Step 2: Solve the equation

Now perform the operations. This gives \(a_{16} = 9 + 15 \times 2\).

Step 3: Calculate the final value

Finally after performing the multiplication and addition, you get \(a_{16} = 39\).

Key Concepts

Common Difference in Arithmetic Sequences

In arithmetic sequences, the **common difference** refers to the consistent amount that separates consecutive terms. This difference is key to ensuring that each term in the sequence follows a predictable pattern. For example, in the problem provided, we have a common difference, denoted by the symbol \(d\), which equals 2. This means that every term in the sequence increases by 2 units from the previous term.

This concept can be identified by:

• Subtracting any term from the following term within the sequence.
• Knowing the first term and the common difference allows us to determine each subsequent term.

Understanding the role of the common difference simplifies the process of predicting later terms in the sequence, as it becomes a repetitive addition of this fixed gap.

The First Term of an Arithmetic Sequence

The **first term** of an arithmetic sequence is often represented by \(a_1\) and serves as the starting point for the sequence. It is the value from which all other terms in the sequence are derived based on the common difference. In the exercise example, the first term, \(a_1\), is given as 9.

Here's how it functions:

• The first term sets the sequence's initial position.
• All subsequent terms are built upon the value of this first term plus increments of the common difference.

By knowing the first term, a specific sequence can be defined, and it forms the basis upon which all calculations and predictions for the future sequence terms are made.

Using the Arithmetic Sequence Formula

The **arithmetic sequence formula** is a simple yet powerful tool to calculate any term in the sequence by applying a straightforward calculation. The general formula for finding the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term, \(d\) represents the common difference, and \(n\) is the term number you need to find. For the exercise, substituting the values \(a_1 = 9\), \(d = 2\), and \(n = 16\), simplifies the formula to \(a_{16} = 9 + (16-1) \, 2\).

This formula allows for:

• Efficient calculation of any term without listing all previous terms.
• Identifying patterns within the sequence by analyzing term positions and values.

Thus, the arithmetic sequence formula is indispensable for efficiently managing and working with arithmetic sequences in various mathematical scenarios.