Problem 38 - 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. $$11,8,5,2, \dots$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Common Difference

In an arithmetic sequence, the common difference can be found by subtracting a term from the previous term. Let's compute the common difference using the first two terms: \[ d = 8 - 11 = -3 \] Thus, the common difference is \(-3\).

Step 2: Find the Fifth Term

Once we have the common difference, we can find any term in the sequence using the formula for the \(n\)th term of an arithmetic sequence: \[ a_n = a_1 + (n-1) imes d \]Let's find the fifth term (\(a_5\)):\[ a_5 = 11 + (5-1) imes (-3) = 11 + 4 \times (-3) = 11 - 12 = -1 \] Therefore, the fifth term is \(-1\).

Step 3: Express the General Form of the Sequence

The formula for the \(n\)th term of the sequence is derived from the arithmetic sequence formula:\[ a_n = a_1 + (n-1) imes d \]Substituting for \(a_1 = 11\) and \(d = -3\):\[ a_n = 11 + (n-1) imes (-3) \]\[ a_n = 11 - 3n + 3 = -3n + 14 \]Thus, the \(n\)th term of the sequence is \(-3n + 14\).

Step 4: Calculate the 100th Term

To find the 100th term, substitute \(n = 100\) into the general formula:\[ a_{100} = -3(100) + 14 \]\[ a_{100} = -300 + 14 = -286 \]Therefore, the 100th term is \(-286\).

Key Concepts

Common Difference

In arithmetic sequences, one of the key components you'll encounter is the **common difference**. It's a crucial element to determine how each term relates to the ones before and after it. The common difference, often denoted as **d**, is simply the result of subtracting one term from the term preceding it.
For example, consider the sequence provided: 11, 8, 5, 2, ...
By calculating the difference between the first two terms:

• Common Difference (d) = 8 - 11 = -3

The common difference here is **-3**. This number tells us that to advance from one term to the next, we subtract 3. Recognizing this pattern of consistent subtraction is fundamental in dealing with arithmetic sequences.
With this understanding, you can predict future terms and better grasp the structure of the sequence.

nth Term Formula

Finding any given term in an arithmetic sequence can be simplified with the **nth term formula**. This valuable formula allows you to pinpoint any term without listing the entire sequence.
The formula for the nth term, represented as \(a_n\), is expressed as:

• \( a_n = a_1 + (n-1) \times d \)

Where **\(a_1\)** is the first term in the sequence, **n** stands for the term number you're looking for, and **d** is the common difference.
Using this formula with our sequence 11, 8, 5, 2,..., we substitute \(a_1 = 11\) (the first term) and \(d = -3\):

• \( a_n = 11 + (n-1)(-3) \)

Simplifying further, you arrive at:

• \( a_n = -3n + 14 \)

This simplified equation can give us any term, such as the fifth or the 100th term, when we substitute the respective value of **n**. Recognizing and applying this formula is a powerful tool in solving arithmetic sequence problems efficiently.

Sequence Pattern

Understanding the **sequence pattern** in arithmetic sequences provides insight into the predictable rhythm and formation of these sequences. Each term in an arithmetic sequence connects through a fixed addition or subtraction described by the common difference.
For our given sequence, you can quickly identify the ongoing pattern: 11, 8, 5, 2,...
Here, each term is 3 less than the preceding term, as previously calculated through the common difference. Recognizing that each term is part of a linear progression can help you:

• Predict future terms easily
• Understand how terms relate to one another
• Apprehend the sequence's overall behavior

For instance, knowing the initial term is 11 and understanding that the pattern involves subtracting 3 each time lets us correctly deduce future terms without manual calculation of each subtraction.
Grasping this logical flow in arithmetic sequences helps strengthen your mathematical intuition and allows you to tackle more complex problems with confidence.