Problem 37 - 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

The common difference is 5. The fifth term is 24. The 100th term is 499.

Step 1: Identify the Common Difference

In an arithmetic sequence, the common difference is the difference between any two consecutive terms. You can find it by subtracting the first term from the second term. Here, the first term is 4 and the second term is 9. So:\[ d = 9 - 4 = 5 \]The common difference of the arithmetic sequence is 5.

Step 2: Calculate the Fifth Term

To find the fifth term of an arithmetic sequence, use the formula for the \(n\)th term, which is \(a_n = a_1 + (n-1) \, d \). Here, \(a_1 = 4\) and \(d = 5\), and you want the fifth term, so \(n = 5\):\[ a_5 = 4 + (5-1) \, 5 = 4 + 4 \, imes \, 5 = 4 + 20 = 24 \]So, the fifth term is 24.

Step 3: Derive the General Formula for the n-th Term

The general formula for the \(n\)th term of an arithmetic sequence is given by \(a_n = a_1 + (n-1) \, d \). For this sequence, the first term \(a_1\) is 4 and the common difference \(d\) is 5. Therefore, the formula is:\[ a_n = 4 + (n-1) \, 5 \]Simplifying this, we get:\[ a_n = 4 + 5n - 5 = 5n - 1 \]

Step 4: Find the 100th Term Using the General Formula

Now that we have the general formula \(a_n = 5n - 1\), substitute \(n = 100\) to find the 100th term:\[ a_{100} = 5(100) - 1 = 500 - 1 = 499 \]Thus, the 100th term of the sequence is 499.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the constant amount that each term increases by to get to the next term. It's like stepping up a staircase where each step is the same height. To find the common difference, you subtract the preceding term from the succeeding term.
Let's look at an example:

• The sequence given is 4, 9, 14, 19, ...
• To find the common difference, subtract the first term from the second term:
• \[ d = 9 - 4 = 5 \]

The common difference here is 5, meaning each number in the sequence increases by 5 to get to the next. Understanding this simple subtraction can help you quickly identify the pattern in arithmetic sequences.

General Term Formula

The general term formula allows us to find any term in an arithmetic sequence without listing all previous terms. This formula is a useful tool since it provides a quick way to jump directly to any term position we want.
The formula for the general term of an arithmetic sequence is:

• \[ a_n = a_1 + (n-1) \, d \]
• Where:
  • \( a_n \) is the term you're looking to find.
  • \( a_1 \) is the first term of the sequence.
  • \( d \) is the common difference.
  • \( n \) is the term’s position in the sequence.

For our sequence, where the first term \( a_1 \) is 4 and the common difference \( d \) is 5, we plug these into the formula:
\[ a_n = 4 + (n-1) \, 5 \]
Simplify it:
\[ a_n = 5n - 1 \]
This simplified formula helps you find the \( n \)-th term directly.

Nth Term Calculation

Calculating the \( n \)-th term of an arithmetic sequence is straightforward with the help of the general term formula. Let's see how to find the 5th term and even the 100th with the earlier formula.

• Start by using the formula: \[ a_n = 5n - 1 \]
• For the 5th term (\( n = 5 \)):
• \[ a_5 = 5(5) - 1 = 25 - 1 = 24 \]

The 5th term is 24. Easy, right? Now, let's calculate a larger term, like the 100th.

• Plug \( n = 100 \) into the formula:
• \[ a_{100} = 5(100) - 1 = 500 - 1 = 499 \]

So, the 100th term is 499. Once you have your formula, you can quickly and easily find any term in the sequence without listing all the terms up to that point.