Problem 41 - 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

Common difference = 1.5; fifth term = 31; nth term = \(1.5n + 23.5\); 100th term = 173.5.

Step 1: Identify the first term and second term

Given the sequence: 25, 26.5, 28, 29.5, ... The first term of the sequence is 25. The second term of the sequence is 26.5.

Step 2: Calculate the common difference

To find the common difference in an arithmetic sequence, subtract the first term from the second term.Common difference, \(d = 26.5 - 25 = 1.5\).

Step 3: Find the formula for the nth term of an arithmetic sequence

The formula for the nth term of an arithmetic sequence is given by:\(a_n = a_1 + (n-1) \cdot d\).Where \(a_1\) is the first term and \(d\) is the common difference.

Step 4: Calculate the fifth term

Substitute \(n = 5\), \(a_1 = 25\), and \(d = 1.5\) into the nth term formula:\(a_5 = 25 + (5-1) \cdot 1.5\).Calculate \(a_5 = 25 + 6 = 31\).So, the fifth term is 31.

Step 5: Find the nth term formula

Substitute \(a_1 = 25\) and \(d = 1.5\) into the nth term formula:\(a_n = 25 + (n-1) \cdot 1.5\).Simplify the equation:\(a_n = 25 + 1.5n - 1.5\),\(a_n = 1.5n + 23.5\).Thus, the nth term of the sequence is \(a_n = 1.5n + 23.5\).

Step 6: Calculate the 100th term

Substitute \(n = 100\) into the nth term formula:\(a_{100} = 1.5 \cdot 100 + 23.5\).Calculate \(a_{100} = 150 + 23.5 = 173.5\).Thus, the 100th term is 173.5.

Key Concepts

Common Difference

In an arithmetic sequence, the **common difference** is the consistent gap or interval between each number in the sequence. To determine the common difference, simply subtract the first term from the second term.
For example, given the sequence 25, 26.5, 28, and 29.5, subtract the first number (25) from the second number (26.5):

• Common difference, \(d = 26.5 - 25 = 1.5\).

This common difference of 1.5 signifies that each term in the sequence increases by 1.5 compared to the previous term. Recognizing and calculating the common difference is crucial because it allows us to understand how the sequence progresses, paving the way for further sequence calculations. As the foundation of an arithmetic sequence, the common difference is used to generate any subsequent terms.

Nth Term Formula

The **nth term formula** is a mathematical expression that helps find any term in an arithmetic sequence without listing all the terms before it. This formula is based on the first term and the common difference, providing a swift method to calculate any term you need.
The formula for the nth term \(a_n\) of an arithmetic sequence is:

• \(a_n = a_1 + (n-1) \cdot d\)

Here, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the position of the term you want to find.
To apply it, for example, if you want to find the nth term for the sequence starting at 25 with a common difference of 1.5, like the sequence from the original problem, substitute these values into the formula:

• \(a_n = 25 + (n-1) \cdot 1.5\)

It helps in easily finding terms, like the 50th or 100th, without manual calculations of each preceding term.

Sequence Calculations

**Sequence calculations** encompass how to use the common difference and nth term formula to find specific terms within an arithmetic sequence. These calculations are invaluable in quickly accessing particular points in the sequence.
To demonstrate, let's calculate the 100th term in our sequence example: We have the nth term formula already determined as \(a_n = 1.5n + 23.5\). To find the 100th term, substitute 100 for \(n\):

• \(a_{100} = 1.5 \cdot 100 + 23.5\)
• Calculate: \(a_{100} = 150 + 23.5 = 173.5\)

Thus, the 100th term is 173.5. These steps highlight how **sequence calculations** make finding terms both easy and systematic, freeing you from repetitive calculations. Once you master the formulas, you can quickly determine even very large or very small terms.