Problem 26 - 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. $$11,8,5,2, \dots$$

Step-by-Step Solution

Answer

The common difference is -3. The fifth term is -1. The nth term is \(14 - 3n\). The 100th term is -286.

Step 1: Identify the Common Difference

An arithmetic sequence has a constant difference between consecutive terms. To find it, subtract the first term from the second term: \(8 - 11 = -3\). Therefore, the common difference, \(d\), is \(-3\).

Step 2: Calculate the Fifth Term

The general formula for the \(n\)th term in an arithmetic sequence is \(a_n = a_1 + (n-1)d\). We need to find \(a_5\), the fifth term. Substitute \(a_1 = 11\), \(d = -3\), and \(n = 5\): \[a_5 = 11 + (5-1)(-3) = 11 - 12 = -1\]. Thus, the fifth term is \(-1\).

Step 3: Determine the General Formula for the nth Term

Use the formula for the \(n\)th term, \(a_n = a_1 + (n-1)d\). Substitute \(a_1 = 11\) and \(d = -3\): \[a_n = 11 + (n-1)(-3) = 11 - 3(n-1)\] This simplifies to: \[a_n = 11 - 3n + 3 = 14 - 3n\]. This is the formula for the \(n\)th term.

Step 4: Calculate the 100th Term

Using the nth term formula \(a_n = 14 - 3n\), substitute \(n = 100\) to find the 100th term: \[a_{100} = 14 - 3(100) = 14 - 300 = -286\]. Thus, the 100th term is \(-286\).

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, the common difference is the distinguishing factor. It's the fixed amount that we add or subtract to the previous term to get the next term in the sequence. This difference is what marks an arithmetic sequence apart from other types of sequences.

To calculate the common difference, we simply subtract the second term from the first term, or any consecutive pair of terms.

• For example, if we have the sequence: 11, 8, 5, 2, ..., the common difference ( d) can be calculated as:\[d = 8 - 11 = -3\]

Hence, for this sequence, each term decreases by 3 from the previous one.

Exploring the nth Term Formula

The nth term formula allows us to find any term in an arithmetic sequence without listing all the terms. This saves time and effort, especially for larger sequences.

The formula for the nth term ( a_n) is:

• \[a_n = a_1 + (n-1)d\]

Here,

• a_n is the nth term,
• a_1 is the first term of the sequence,
• n is the term number,
• d is the common difference.

For the sequence starting with 11 and having a common difference of -3, the nth term formula becomes:

• \[a_n = 11 + (n-1)(-3) = 14 - 3n\]

This formula can help find any term in the sequence efficiently.

Performing Sequence Calculations

To perform sequence calculations, first know the common difference and the formula for the nth term. This knowledge allows us to tackle various problems related to arithmetic sequences.

For example, calculating specific terms, like the fifth term, simply involves:

• Substituting the term number into the nth term formula.

Consider the sequence: 11, 8, 5, 2, ...:

• Let's find the fifth term: For n = 5, use the nth term formula:\[a_5 = 14 - 3 imes 5 = 14 - 15 = -1\]

Thus, the fifth term is -1. You can use similar steps for any other term.

Understanding Arithmetic Progression

Arithmetic progression (AP) involves a sequence of numbers in which each term after the first is derived by adding a constant known as the common difference. This is a straightforward yet powerful mathematical concept.

An AP doesn't change as you move along the sequence; the pattern remains steady:

• The initial term ( a_1) starts the sequence.
• The common difference ( d) drives the progression.
• Every subsequent term can be easily found using the nth term formula:\[a_n = a_1 + (n-1)d\]

Arithmetic progressions are highly useful in real-world scenarios, such as predicting sequences and modeling linear growth patterns, making them an essential concept in both academic studies and practical applications.