Problem 33 - Precalculus Mathematics for Calculus

Question

Determine the common difference, the fifth term, the \(n\) th term, and the 100 th term of the arithmetic sequence. $$25,26.5,28,29.5, \dots$$

Step-by-Step Solution

Answer

The common difference is 1.5. The fifth term is 31, and the 100th term is 173.5.

Step 1: Identify the Common Difference

An arithmetic sequence has a common difference between consecutive terms. To find this difference, subtract the first term from the second term: \(d = 26.5 - 25 = 1.5\). Therefore, the common difference is \(d = 1.5\).

Step 2: Find the Fifth Term

To find the fifth term, use the formula for the \(n\)th term of an arithmetic sequence: \(a_n = a_1 + (n-1) \cdot d\). For the fifth term \(a_5\): \(a_5 = 25 + (5-1) \cdot 1.5 = 25 + 6 = 31\).So, the fifth term is 31.

Step 3: Find the General \(n\)th Term

The general formula for the \(n\)th term of the sequence is given by \(a_n = a_1 + (n-1) \cdot d\). Using the values \(a_1 = 25\) and \(d = 1.5\), the formula becomes: \(a_n = 25 + (n-1) \cdot 1.5\). Simplifying gives \(a_n = 1.5n + 23.5\).

Step 4: Find the 100th Term

Using the formula for the \(n\)th term: \(a_n = 1.5n + 23.5\), substitute \(n = 100\):\(a_{100} = 1.5 \cdot 100 + 23.5 = 150 + 23.5 = 173.5\). Thus, the 100th term is 173.5.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key component. It is the consistent difference between any two consecutive terms in the sequence. For instance, in the sequence 25, 26.5, 28, 29.5, ..., we identify the common difference, denoted as \(d\), by subtracting the first term from the second: \(d = 26.5 - 25 = 1.5\).

Knowing the common difference helps us predict the next terms in the sequence. We can continually add this value to the last known term to find subsequent terms. This rules out any guesswork and establishes a pattern.

n-th Term Formula

The formula for determining the \(n\)-th term of an arithmetic sequence is cleverly constructed to help us find any term at a specific position \(n\). The formula is \(a_n = a_1 + (n-1) \cdot d\), where \(a_n\) is the \(n\)-th term, \(a_1\) is the first term, and \(d\) is the common difference.

In our example sequence, 25, 26.5, 28, 29.5, ..., \(a_1 = 25\) and \(d = 1.5\). So, the \(n\)-th term is calculated as \(a_n = 25 + (n-1) \cdot 1.5\). This can be simplified to \(a_n = 1.5n + 23.5\).

This formula is powerful because it allows us to find any term in the sequence without having to list all the terms before it.

Sequence Terms

Sequence terms in an arithmetic sequence follow a distinct pattern determined by the common difference. After establishing the first term and the common difference, all subsequent terms can be determined with ease.

• The first term, \(a_1\), acts as a starting point. In our example, it is 25.
• Each following term is the sum of the previous term and the common difference (1.5 in our example). This incremental pattern continues indefinitely.
• For instance, to find the fifth term, we apply the formula: \(a_5 = 25 + (5-1) \cdot 1.5 = 31\).

Understanding sequence terms is crucial as it provides a clear path for predicting values further down the line, such as the 100th term in our example, which is calculated to be 173.5 using the \(n\)-th term formula.