Problem 15 - Algebra 2

Question

Find the 32nd term of each sequence. \(101,105,109,113, \ldots\)

Step-by-Step Solution

Answer

The 32nd term of the sequence is 225.

Step 1: Identify the first term and common difference

The first number in the sequence is \( a_1 = 101 \). It is noticed from the given numbers that they are increasing by 4 each time. So, the common difference \( d = 4 \).

Step 2: Plug the values into the formula

For the formula \( a_n = a_1 + (n - 1) * d \), replace \( a_1 \) with 101, \( n \) with 32 (since the 32nd term is being sought), and \( d \) with 4. The resulting formula is \( a_{32} = 101 + (32 - 1) * 4 \).

Step 3: Calculate the 32nd term

Now complete the calculation: 101 + 31 * 4 = 101 + 124 = 225.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the constant amount that each term increases or decreases by. This difference is crucial in identifying how the sequence progresses. For example, in the sequence provided: 101, 105, 109, 113, ..., each term increases by 4.
This number, 4, is our common difference. The common difference can be calculated by subtracting any term from the term that follows it.

• Example: In the sequence, subtract the first term from the second: 105 - 101 = 4.
• This arithmetic sequence can be represented as having a common difference of 4.

Understanding the common difference allows us to predict future terms in the sequence. It's consistent across the entire sequence, which simplifies calculations when using the sequence formula.

Sequence Formula

An arithmetic sequence is mathematically represented by a specific formula. This sequence formula is essential for finding any term within an arithmetic sequence without listing all the terms.
The formula is:

• \[a_n = a_1 + (n - 1) \cdot d\]
• \(a_n\) is the nth term of the sequence you're looking for.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference.
• \(n\) is the term number.

This formula allows you to bypass manually calculating each term to reach your desired term in the sequence, greatly simplifying the process.

Term Calculation

Calculating a specific term in an arithmetic sequence involves substituting known values into the sequence formula. Let's apply this to find the 32nd term of the sequence given in the exercise:

• First, identify the known elements:
• \(a_1 = 101\), the first term.
• \(d = 4\), the common difference.
• \(n = 32\), the term number of interest.

Insert these values into the sequence formula:

• \[a_{32} = 101 + (32 - 1) \cdot 4\]
• Calculate inside the parentheses first: \(32 - 1 = 31\).
• Multiply by the common difference: \(31 \cdot 4 = 124\).
• Add this to the first term: \(101 + 124 = 225\).

The 32nd term in the sequence is 225. By using the formula, finding any term becomes a straightforward process, regardless of its position in the sequence.