Problem 29 - 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. $$4,9,14,19, \dots$$

Step-by-Step Solution

Answer

Common difference: 5; Fifth term: 24; \(n\)th term: \(5n-1\); 100th term: 499.

Step 1: Identify the first term

The first term of the arithmetic sequence is the first number in the list. Here, the first term \(a_1\) is 4.

Step 2: Determine the common difference

In an arithmetic sequence, the common difference \(d\) is the difference between any two successive terms. Calculate \(9 - 4\) to get 5. This is the common difference \(d = 5\).

Step 3: Calculate the fifth term

To find the fifth term \(a_5\), use the formula \(a_n = a_1 + (n-1) \, d\). Substituting the known values, \(a_5 = 4 + (5-1) \, 5 = 4 + 20 = 24\).

Step 4: Express the \(n\)th term formula

The \(n\)th term of an arithmetic sequence can be calculated using the formula \(a_n = a_1 + (n-1) \, d\). Substituting \(a_1 = 4\) and \(d = 5\), we get \(a_n = 4 + (n-1) \, 5\). Simplifying, \(a_n = 5n - 1\).

Step 5: Calculate the 100th term

Substitute \(n = 100\) into the \(n\)th term formula \(a_n = 5n - 1\). So, \(a_{100} = 5 \times 100 - 1 = 500 - 1 = 499\).

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, the term 'common difference' refers to the consistent difference between consecutive terms. Identifying this is crucial, as it helps in predicting future numbers in the sequence.
For our given sequence, the numbers are 4, 9, 14, 19, and so on. To find the common difference, simply subtract any term from the subsequent term. Here, the calculation is done as follows:

• Subtract the first term from the second: 9 - 4 = 5
• Subtract the second term from the third: 14 - 9 = 5
• Subtract the third term from the fourth: 19 - 14 = 5

The consistent result shows that our common difference, denoted by \(d\), is 5.
Understanding this difference is essential, as it helps in forming predictions and calculating any term in the arithmetic sequence.

Exploring the nth Term Formula

To find any term in an arithmetic sequence, we use the 'nth term' formula. This is a straightforward formula that helps calculate the value of any term given its position \(n\).
The general formula for the nth term is: \[a_n = a_1 + (n-1) \times d\]Here,

• \(a_n\) is the nth term
• \(a_1\) is the first term, which is 4 in our sequence
• \(d\), the common difference, is 5

Let's put the values into the formula:\[a_n = 4 + (n-1) \times 5\]Simplified, becomes:\[a_n = 5n - 1\]This formula can then be used to find any term in the sequence. It demonstrates how arithmetic sequences have a linear structure, making them predictable.

Finding the Fifth Term

Finding a specific term in an arithmetic sequence can be easily done using the nth term formula. In this instance, we are looking for the fifth term \(a_5\).
Using the nth term formula:\[a_n = a_1 + (n-1) \times d\]Substitute \(n = 5\):\(a_5 = 4 + (5-1) \times 5\)Breaking it down:

• Start with \(n-1\): \(5 - 1 = 4\)
• Multiply by the common difference \(d = 5\): \(4 \times 5 = 20\)
• Add this to the first term \(a_1 = 4\): \(4 + 20 = 24\)

Thus, the fifth term \(a_5\) is calculated to be 24.
This method stands for all other specific terms, given the formula and common difference, allowing easy computation for any position in the sequence.