Problem 28 - Algebra and Trigonometry

Question

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ 1,5,9,13, \dots $$

Step-by-Step Solution

Answer

Common difference: 4, fifth term: 17, nth term: \( 4n - 3 \), 100th term: 397.

Step 1: Identify the common difference

In an arithmetic sequence, the common difference is the amount each term increases by. To find it, subtract the first term from the second term: \[ d = 5 - 1 = 4 \] Thus, the common difference is 4.

Step 2: Calculate the fifth term

To find the fifth term of the sequence, use the formula for the nth term of an arithmetic sequence: \[ a_n = a_1 + (n-1) imes d \] Where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference. For the fifth term: \[ a_5 = 1 + (5-1) imes 4 = 1 + 16 = 17 \] So, the fifth term is 17.

Step 3: Derive the nth term formula

Using the same nth term formula \( a_n = a_1 + (n-1) imes d \), substitute \( a_1 = 1 \) and \( d = 4 \) to find the general formula: \[ a_n = 1 + (n-1) imes 4 = 1 + 4n - 4 = 4n - 3 \] So, the nth term formula is \( a_n = 4n - 3 \).

Step 4: Find the 100th term

Using the nth term formula \( a_n = 4n - 3 \), substitute \( n = 100 \) to find the 100th term: \[ a_{100} = 4 imes 100 - 3 = 400 - 3 = 397 \] Therefore, the 100th term is 397.

Key Concepts

Common Difference

The concept of a common difference is crucial to understanding arithmetic sequences. This difference is the constant value that separates each number from the next in the sequence. To find the common difference, take any term in the sequence and subtract from it the preceding term.
For example, in the sequence given: 1, 5, 9, 13, ..., you take the second term (5) and subtract the first term (1):

• \( d = 5 - 1 = 4 \)

This result means that each term in the sequence is 4 more than the one before it. Understanding the common difference helps in predicting future terms and understanding the pattern that the sequence follows.

Nth Term

Finding the nth term in an arithmetic sequence means expressing the formula that can calculate any term in the sequence using its position number. The nth term formula for an arithmetic sequence is given by:

• \( a_n = a_1 + (n-1) \times d \)

Here, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference. For our sequence, substituting \( a_1 = 1 \) and \( d = 4 \), we have:

• \( a_n = 1 + (n-1) \times 4 = 4n - 3 \)

This formula reveals the pattern in the sequence. As you change \( n \), you directly find the corresponding term in the sequence.

Fifth Term

Calculating the fifth term involves using the nth term formula to find a particular specified term in the sequence. To find the fifth term, substitute \( n = 5 \) into the nth term formula:

• \( a_5 = 4 \times 5 - 3 \)
• \( a_5 = 20 - 3 = 17 \)

Therefore, the fifth term in the sequence is 17. The formula allows you to find this term efficiently without needing to list all previous terms manually.

100th Term

The 100th term can be found with the same nth term formula, demonstrating the power and utility of understanding arithmetic sequences. Substitute \( n = 100 \) into the formula to find the 100th term:

• \( a_{100} = 4 \times 100 - 3 \)
• \( a_{100} = 400 - 3 = 397 \)

This method allows you to directly jump to any term in the sequence without having to calculate or write every preceding term, saving time and reducing errors.