Problem 45 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{2}=5, a_{6}=13$$

Step-by-Step Solution

Answer

The general term is \(a_n = 2n + 1\).

Step 1: Identify the known values

In the arithmetic sequence, we know that the second term \(a_2 = 5\) and the sixth term \(a_6 = 13\). We need to use these known values to find the common difference \(d\) and the first term \(a_1\).

Step 2: Use the formula for terms in an arithmetic sequence

The general formula for any term in an arithmetic sequence is \(a_n = a_1 + (n-1)d\). Using this formula, we can set up two equations based on the given terms:\(a_2 = a_1 + d = 5\) and \(a_6 = a_1 + 5d = 13\).

Step 3: Solve the system of equations

We have the system of equations: \(a_1 + d = 5\) and \(a_1 + 5d = 13\). Subtract the first equation from the second: \((a_1 + 5d) - (a_1 + d) = 13 - 5\), which simplifies to \(4d = 8\). Solve for \(d\) to get \(d = 2\).

Step 4: Find the first term \(a_1\)

Substitute \(d = 2\) back into the first equation \(a_1 + d = 5\). Thus, \(a_1 + 2 = 5\), which simplifies to \(a_1 = 3\).

Step 5: Construct the general term \(a_n\)

Now that we have \(a_1 = 3\) and \(d = 2\), substitute these values back into the general term formula: \(a_n = a_1 + (n-1)d\). Therefore, the general term for the sequence is \(a_n = 3 + (n-1) \times 2\). Simplify it to \(a_n = 3 + 2n - 2 = 2n + 1\).

Key Concepts

Common Difference

The common difference in an arithmetic sequence is crucial as it defines the structure and pattern of the sequence. It is the amount by which each term in the sequence increases or decreases as you move from one term to the next. The common difference, often denoted by \(d\), is calculated by subtracting any term in the sequence from the subsequent term.
For instance, in the sequence where we know that \(a_2 = 5\) and \(a_6 = 13\), we can use the formula previously derived:

• \(a_6 = a_1 + 5d = 13\)
• \(a_2 = a_1 + d = 5\)

By solving these equations together, we simplify to get \(4d = 8\), leading us to \(d = 2\). This common difference \(d = 2\) ensures that each term of the sequence increases by 2 as you progress from one term to the next. Recognizing the common difference allows for the prediction and construction of future terms beyond the ones given.

First Term of Sequence

The first term of the arithmetic sequence, labeled \(a_1\), serves as the starting point from which the sequence begins and provides a base for generating all remaining terms. Once the common difference \(d\) is known, the first term \(a_1\) can be easily found by rearranging one of the sequence formulas.
In our given sequence, we worked with the equation \(a_1 + d = 5\), substituting \(d = 2\), yielding \(a_1 + 2 = 5\). Solving this equation finds \(a_1 = 3\). This result indicates that the sequence starts at 3, and each subsequent term is calculated by adding the common difference to this initial value.
Having \(a_1\) clearly laid out allows you to ascertain the sequence from its foundation step by step.

General Term Formula

The general term formula, denoted as \(a_n\), provides a comprehensive rule for finding any specific term within an arithmetic sequence. It is derived from the understanding of both the first term and the common difference. The formula is expressed as:

• \(a_n = a_1 + (n-1)d\)

Here, \(a_1\) is the first term, \(n\) represents the position of the term in the sequence, and \(d\) is the common difference.
By substituting \(a_1 = 3\) and \(d = 2\) from our example, we adapt the formula to become \(a_n = 3 + (n-1) \times 2\). Simplifying yields \(a_n = 2n + 1\). This equation succinctly defines a means to compute any term's value in the sequence by simply plugging in the desired term's position \(n\).
Mastery of the general term formula empowers you to explore and create terms scattered any number of steps ahead in an arithmetic sequence without starting from the first term.