Problem 20 - Algebra 2

Question

Write an explicit formula for each sequence. Then find \(a_{12}\) $$ 4,7,10,13,16, \dots $$

Step-by-Step Solution

Answer

The explicit formula for the sequence is \(a_n = 3n + 1\). The 12th term of the sequence is 37.

Step 1: Identify the common difference

In an arithmetic sequence, the difference between successive terms is constant. This is known as the common difference, denoted by \(d\). By subtracting the first term from the second term in the sequence, 7 - 4 = 3, we identify the common difference \(d\) to be 3.

Step 2: Write the explicit formula

Once we've found the common difference, we can write the explicit formula for the sequence. From the formula \(a_n = a_1 + (n-1) * d\), we know that \(a_1\) is the first term of the sequence, which is 4 and \(d\) is the common difference, which is 3. So, substituting these values, the explicit formula is \(a_n = 4 + (n-1) * 3\), that simplifies to \(a_n = 3n + 1\).

Step 3: Find \(a_{12}\)

To find the 12th term in the sequence, we substitute \(n = 12\) into the explicit formula. So, \(a_{12} = 3*12 + 1 = 36 + 1 = 37\).

Key Concepts

explicit formula

Every arithmetic sequence can be represented using an explicit formula. An explicit formula allows us to calculate any term in the sequence directly without finding all the previous terms. This makes it a powerful tool, especially when we need to find terms that are far along in the sequence.
In the context of an arithmetic sequence, an explicit formula is generally given by:

• The starting term, denoted as \(a_1\), which is the first number in the sequence.
• The common difference \(d\), indicating how much each term increases by compared to the previous one.

Using these, the explicit formula is written as \(a_n = a_1 + (n-1) \times d\).
For the sequence 4, 7, 10, 13, 16,..., we have \(a_1 = 4\) and \(d = 3\). Thus, the explicit formula becomes \(a_n = 4 + (n-1)\times 3\). Simplifying this results in \(a_n = 3n + 1\). This formula allows you to easily calculate any desired term in the sequence.

common difference

In an arithmetic sequence, each term after the first is calculated by adding a fixed number, known as the common difference \(d\), to the previous term.
This consistent addition is what distinguishes an arithmetic sequence from other types of sequences.
In the given exercise sequence: 4, 7, 10, 13, 16,..., discovering the common difference involves:

• Taking any two consecutive terms and subtracting the first term from the second.
• For example, 7 - 4 yields 3.

This subtraction confirms that for this sequence, the common difference \(d\) is 3.
Thus, each term can be derived from the previous term by adding the common difference.

sequence terms

The terms of a sequence are the individual elements that make up the sequence. They are crucial in understanding the pattern and behavior of the sequence.
In arithmetic sequences, each term follows a regular pattern established by the common difference. Using the explicit formula, not only can we find these terms, but we can understand their linear distribution across the sequence.
For example, in the sequence: 4, 7, 10, 13, 16,..., each of these numbers is a term. To find the nth term, we use the explicit formula \(a_n = 3n + 1\).
For instance, to find the 12th term \(a_{12}\), we substitute \(n = 12\) into the formula:

• \(a_{12} = 3 \times 12 + 1 = 36 + 1 = 37\).

As seen, the explicit formula offers a streamlined way to identify specific terms within a sequence, illustrating the predictable nature of arithmetic sequences.