Problem 29 - Precalculus: Mathematical 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, the nth term formula is \(a_n = 25 + (n-1) \cdot 1.5\), and the 100th term is 173.5.

Step 1: Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is the same. This difference is known as the common difference.

Step 2: Find the Common Difference

To find the common difference of the sequence, subtract the first term from the second term. The first term is 25 and the second term is 26.5. \[ Common \ Difference = 26.5 - 25 = 1.5 \]

Step 3: Determine the Fifth Term

To find the fifth term, use the formula for the nth term of an arithmetic sequence: \[ a_n = a_1 + (n-1) \cdot d \]where \( a_1 \) is the first term (25), \( d \) is the common difference (1.5), and \( n \) is the term number (5).Substitute the values into the formula: \[ a_5 = 25 + (5-1) \cdot 1.5 = 25 + 6 = 31 \]

Step 4: Find the General Formula for the nth Term

The general formula to find the nth term of an arithmetic sequence is given by:\[ a_n = a_1 + (n-1) \cdot d \]Substitute \( a_1 = 25 \) and \( d = 1.5 \) into the formula to find the nth term:\[ a_n = 25 + (n-1) \cdot 1.5 \]

Step 5: Determine the 100th Term

Use the nth term formula to find the 100th term by substituting \( n = 100 \):\[ a_{100} = 25 + (100-1) \cdot 1.5 = 25 + 148.5 = 173.5 \]

Key Concepts

Common Difference

In an arithmetic sequence, the term "common difference" refers to the constant difference between any two consecutive terms. It remains the same throughout the sequence. This means if you subtract any term from the next one, you always get the common difference.

To find the common difference in our specific sequence, we subtract the first term from the second term. Here, the first term is 25, and the second term is 26.5. Therefore, the common difference is calculated as:

• Common Difference = 26.5 - 25 = 1.5

Thus, each term in the sequence increases by 1.5 more than the previous term.

nth Term Formula

The nth term of an arithmetic sequence can be found using a simple formula. This formula allows you to calculate any term in the sequence without listing all the terms up to that point.

The formula for the nth term is:

• \( a_n = a_1 + (n-1) \cdot d \)

Where:

• \( a_1 \) is the first term of the sequence,
• \( n \) is the position of the term you want to find,
• And \( d \) is the common difference.

For our sequence, we substitute \( a_1 = 25 \) and \( d = 1.5 \) to get the nth term formula:

• \( a_n = 25 + (n-1) \cdot 1.5 \)

This is a fantastic tool for quickly finding any term in the sequence!

Fifth Term

To find the fifth term in the sequence, we use the nth term formula. Here, \( n = 5 \) because we're interested in the fifth position.

Using the formula:

• \( a_5 = 25 + (5-1) \cdot 1.5 \)

Breaking it down:

• \( a_5 = 25 + 4 \cdot 1.5 \)
• \( a_5 = 25 + 6 \)
• \( a_5 = 31 \)

So, the fifth term in this sequence is 31. Each time you see a sequence, remember the formula helps you jump to any term directly.

100th Term

The 100th term of the sequence can also be easily found using the nth term formula. Here, you substitute \( n = 100 \) into the formula.

Let's apply it and see what happens:

• \( a_{100} = 25 + (100-1) \cdot 1.5 \)

This breaks down to:

• \( a_{100} = 25 + 99 \cdot 1.5 \)
• \( a_{100} = 25 + 148.5 \)
• \( a_{100} = 173.5 \)

Therefore, the 100th term in the sequence is 173.5. With the nth term formula, you can find any term, no matter how far along in the sequence, without trouble.