Problem 22 - Prealgebra

Question

Write an equation that describes each sequence. Then find the indicated term. \(16,17,18,19, \dots ; 23 \mathrm{rd}\) term

Step-by-Step Solution

Answer

The 23rd term is 38.

Step 1: Identify the Sequence Type

The sequence given is \(16, 17, 18, 19, \ldots \). Notice that each term increases by 1, indicating it is an arithmetic sequence.

Step 2: Determine the First Term and Common Difference

In the arithmetic sequence, the first term (\(a\)) is 16, and the common difference (\(d\)) is 1, since each term increases by 1.

Step 3: Write the General Formula for the Arithmetic Sequence

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a + (n-1) \, d\). Substituting the known values, we get \(a_n = 16 + (n-1) \, 1 = 16 + n - 1 = n + 15\).

Step 4: Substitute to Find the 23rd Term

Use the formula \(a_n = n + 15\) to find the 23rd term by substituting \(n = 23\). Thus, \(a_{23} = 23 + 15 = 38\).

Key Concepts

Common Difference in Arithmetic Sequences

In any arithmetic sequence, finding the common difference helps you understand the pattern of how numbers are arranged. This difference is what remains consistent by which each term increases or decreases from the previous one. In the given sequence

• By looking at the numbers 16, 17, 18, and so on, we can tell that each number increases by 1.
• This constant change between consecutive terms is known as the common difference.
• For our specific sequence, the common difference, denoted by \(d\), is \(1\).

Understanding the common difference is key to writing formulas and predicting future numbers in the sequence.

The General Formula for Arithmetic Sequences

The general formula for an arithmetic sequence helps us find any term in the sequence without listing all the terms first. This formula is expressed as:\[a_n = a + (n-1) \, d\]where:

• \(a_n\) is the \(n\)-th term you want to find,
• \(a\) is the first term of the sequence,
• \(n\) represents the term number,
• \(d\) is the common difference.

For our sequence starting with 16 and increasing by 1, the general formula becomes:

• Substituting the values into the formula gives \(a_n = 16 + (n-1) \times 1 = n + 15\).

This equation makes it simple to calculate any term without listing every number.

Finding the N-th Term in an Arithmetic Sequence

Once you have the general formula, finding the \(n\)-th term is straightforward. Whether you're looking for the 23rd term or the 100th, the approach is simple. Let's break it down for the 23rd term:

• We know the formula is \(a_n = n + 15\).
• To find the 23rd term, we replace \(n\) with 23.
• This gives us \(a_{23} = 23 + 15 = 38\).

Understanding how to substitute \(n\) in the general formula allows you to find any term quickly and accurately, regardless of its position in the sequence.

Understanding Arithmetic Progressions

Arithmetic progressions, or arithmetic sequences, are common in mathematics and beyond. They form a linear pattern where the difference between each consecutive term is constant.

• The sequence \(16, 17, 18, ...\) is a clear example of an arithmetic progression with a common difference of 1.
• These sequences can be used in various real-world applications, such as calculating loan repayments or predicting patterns in data.
• The ability to analyze and find patterns in arithmetic sequences helps in developing problem-solving skills.

Grasping the concept of arithmetic progressions provides a foundation for further exploration into more complex mathematical topics and sequences.