Problem 15 - 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 \(a_{6}\) when \(a_{1}=13, d=4\)

Step-by-Step Solution

Answer

The 6th term of the arithmetic sequence, \(a_6\), is 33

Step 1: Identify the given parameters

Our given parameters are \(a_1 = 13\), \(d = 4\), and \(n = 6\), which represents the position of the term we are trying to find.

Step 2: Plug into the formula

We need to substitute \(a_1\), \(d\) and \(n\) into the arithmetic sequence formula \(a_n = a_1 + (n-1)d\). Therefore, the equation becomes \(a_6 = a_1 + (6-1) * d\).

Step 3: Calculation

By plugging into the values of \(a_1\) and \(d\) we get, \(a_6 = 13 + (6-1) * 4\). Simplifying the calculation, we get \(a_6 = 13 + 20 = 33\).

Key Concepts

Arithmetic Sequence Formula

Understanding the arithmetic sequence formula is essential for solving problems related to arithmetic sequences. An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant, known as the common difference, to the previous term.

The formula to find the n-th term of an arithmetic sequence is expressed as:
\[ a_n = a_1 + (n-1)d \]
Where:

• \(a_n\) is the n-th term of the sequence,
• \(a_1\) is the first term of the sequence,
• \(n\) is the term number, and
• \(d\) is the common difference between the terms.

By plugging in the appropriate values into this formula, one can calculate any term in an arithmetic sequence. This formula is powerful because it allows you to directly find any term without having to write out the entire sequence.

Common Difference

The common difference in an arithmetic sequence is what sets it apart from other types of sequences. It is the constant value added to each term to get the next term in the sequence. In other words, it is the difference between any two consecutive terms.

For example, in an arithmetic sequence where the first term is 3 and the second term is 7, the common difference \(d\) is 4, because \(7-3=4\). That would mean the sequence is progressing as 3, 7, 11, 15, and so on, with 4 being systematically added to reach the next term.

Identifying the common difference is a crucial step in solving sequence problems, as it allows us to apply the arithmetic sequence formula effectively. In the given exercise, the common difference is \(d=4\), which indicates that every term in the sequence is 4 units apart from its adjacent terms.

Sequence Term Calculation

Calculating a specific term in an arithmetic sequence involves using the arithmetic sequence formula by substituting the value of the first term, the common difference, and the position number of the term we want to calculate.

In the given problem, we are asked to find the 6th term (\(a_6\)) in the sequence where the first term is \(a_1=13\) and the common difference \(d=4\). Following the arithmetic sequence formula, we substitute the values as follows:
\[ a_6 = a_1 + (6-1)\cdot d \]
Simplifying further, the calculation would be:
\[ a_6 = 13 + (6-1)\cdot4 = 33 \]
The 6th term in this sequence is 33. Through this method, we can ascertain any term in the sequence by adjusting our n value as needed. This type of direct computation eliminates the need to write out the entire sequence, which can be particularly helpful for finding terms deep within lengthy sequences.