Problem 44 - 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. $$-t,-t+3,-t+6,-t+9, \dots$$

Step-by-Step Solution

Answer

Common difference: 3; Fifth term: \(-t + 12\); n-th term: \(-t + 3n - 3\); 100th term: \(-t + 297\).

Step 1: Identify the Sequence Pattern

Observe the given sequence: \(-t, -t+3, -t+6, -t+9, \ldots\). Notice that each term is obtained by adding 3 to the previous term. This indicates that the sequence is arithmetic with a common difference.

Step 2: Determine the Common Difference

In an arithmetic sequence, the common difference \(d\) is the difference between any two successive terms. For this sequence: \((-t+3) - (-t) = 3\). Thus, the common difference \(d = 3\).

Step 3: Find the Fifth Term

The first term \(a_1\) is \(-t\). Add the common difference four times to find the fifth term: \(-t + 4\times 3 = -t + 12\). Hence, the fifth term is \(-t + 12\).

Step 4: Establish the General Formula for the n-th Term

The formula for the \(n\)-th term in an arithmetic sequence is \(a_n = a_1 + (n-1)d\). Substituting \(a_1 = -t\) and \(d = 3\), the \(n\)-th term is: \[-t + (n-1)\times 3 = -t + 3n - 3.\] So, the general formula is \(a_n = -t + 3n - 3\).

Step 5: Calculate the 100th Term

Use the general formula for the \(n\)-th term to find the 100th term by substituting \(n = 100\): \[a_{100} = -t + 3(100) - 3 = -t + 300 - 3 = -t + 297.\] Therefore, the 100th term is \(-t + 297\).

Key Concepts

Common Difference

In an arithmetic sequence, the **common difference** is a crucial element. It represents the difference between any two consecutive terms. By identifying this, you can determine the overall pattern of the sequence.
For instance, consider the sequence given in the original exercise:

• each term is written as i.e., the first term, the second term is ( ) - ( ). This simplifies to , showing that the common difference .

This constant amount, which can be positive, negative, or zero, affects how the sequence progresses as you move from one term to the next. If you understand the common difference, you'll unlock the key to exploring and predicting the sequence's behavior.

n-th Term Formula

The **n-th term formula** allows you to find any term in an arithmetic sequence without needing to list all previous terms. This formula is a mathematical expression, providing an efficient means to compute any specific term directly.
For an arithmetic sequence, the formula is given by:

• \[a_n = a_1 + (n-1)d\]

Let's break this down:

• \(a_n\) is the n-th term you want to find.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference.
• \(n\) refers to the term number.

By substituting the known values from our example, such as \(a_1 = -t\) and \(d = 3\), you can quickly find any term such as the fifth or hundredth without laborious calculations. Understanding this formula reduces the complexity of finding terms in large sequences.

Arithmetic Sequence Terminology

Arithmetic sequences have specific terminology essential for grasping the full concept effectively.

• The **first term** (\(a_1\)) is the initial element in the sequence and serves as a starting point for further terms.
• The **common difference** is consistently added to each term to arrive at the next. It's a defining property.
• The **n-th term** formula provides a method to find any term in the sequence promptly without listing all predecessors.
• An **arithmetic sequence** is characterized by this constant difference between consecutive terms, differentiating it from other types of sequences.

With a solid grasp of these terms, you can navigate arithmetic sequences with confidence, applying these concepts across various mathematics problems you may encounter.