Problem 25 - Algebra and Trigonometry

Question

23-32 me the common difference, the fifth term, the \(n\)th term, and the 100th term of the arithmetic sequence. $$4,9,14,19, \dots$$

Step-by-Step Solution

Answer

Common difference is 5; fifth term is 24; nth term formula is \(a_n = 4 + (n-1) \cdot 5\); 100th term is 499.

Step 1: Identify the First Term

The first term of the sequence is given as 4.

Step 2: Find the Common Difference

To find the common difference, subtract the first term from the second term: \(9 - 4 = 5\). Thus, the common difference \(d = 5\).

Step 3: Calculate the Fifth Term

The fifth term of an arithmetic sequence can be found using the formula \(a_n = a_1 + (n-1) \cdot d\). So, \(a_5 = 4 + (5-1) \cdot 5 = 4 + 20 = 24\). The fifth term is 24.

Step 4: Write the Formula for the \(n\)th Term

The formula for the \(n\)th term of an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\). Substituting \(a_1 = 4\) and \(d = 5\), the formula becomes \(a_n = 4 + (n-1) \cdot 5\).

Step 5: Calculate the 100th Term

Using the formula for the \(n\)th term, calculate \(a_{100} = 4 + (100-1) \cdot 5 = 4 + 495 = 499\). The 100th term is 499.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key characteristic. It tells us how much each term in the sequence increases or decreases compared to the previous term. To find the common difference, subtract any term from the term that follows it. Here's how you can do it using the provided sequence:

• The first term is given as 4, and the second term is 9.
• Subtract the first term from the second term: \(9 - 4 = 5\).
• Therefore, the common difference \(d\) is 5.

This common difference remains constant throughout the sequence, which means every term is increased by 5 from the previous one.

Nth Term Formula

The nth term formula is essential for finding any term in an arithmetic sequence. It helps you to calculate terms without listing them all out. The formula is derived as follows:

The first term is denoted as \(a_1\), and the common difference as \(d\). The formula for the nth term, \(a_n\), is:
\[ a_n = a_1 + (n-1) \cdot d \]
To put this into practice with our sequence where \(a_1 = 4\) and \(d = 5\), the nth term formula becomes:
\[ a_n = 4 + (n-1) \cdot 5 \]
This expression makes it straightforward to find any term by simply plugging in the value of \(n\). If you want to find the 5th term, replace \(n\) with 5, and calculate as shown in the step-by-step solution.

Terms of Sequence

Terms of a sequence are essentially the numbers that make up the sequence. Understanding how to identify them and calculate specific ones is fundamental.

• The sequence given starts with the first term \(a_1 = 4\).
• Using the common difference of 5, the terms increase sequentially: 9, 14, 19, and so on.
• Each term is found by adding the common difference to the previous term.

By applying the nth term formula, \(a_n = a_1 + (n-1) \cdot d\), any term like the 100th term can be calculated without listing all the terms. For example, to find the 100th term:
Plug \(n = 100\) into the formula:
\[ a_{100} = 4 + (100-1) \cdot 5 = 499\]
Hence, the 100th term is 499. Knowing how to find these terms is incredibly useful for navigating arithmetic sequences efficiently.