Problem 36 - 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

The common difference is 3; fifth term is \(-t + 12\); n-th term is \(a_n = -t + 3n - 3\); 100th term is \(-t + 297\).

Step 1: Identify the Sequence Structure

The given sequence is an arithmetic sequence. An arithmetic sequence is characterized by a constant difference between consecutive terms. The terms given are \(-t, -t+3, -t+6, -t+9, \dots\).

Step 2: Calculate the Common Difference

To find the common difference \(d\), subtract the first term from the second term: \(-t+3 - (-t) = 3\). Hence, the common difference \(d = 3\).

Step 3: Determine the General Formula for the n-th Term

In an arithmetic sequence, the n-th term \(a_n\) is given by \(a_n = a_1 + (n-1)d\). Here, the first term \(a_1 = -t\) and the common difference \(d = 3\). Substitute these into the formula to get: \(a_n = -t + (n-1) imes 3\). Simplify it to: \(a_n = -t + 3n - 3\).

Step 4: Find the Fifth Term

Substitute \(n = 5\) in the n-th term formula: \(a_5 = -t + 3(5) - 3 = -t + 15 - 3 = -t + 12\). Thus, the fifth term is \(-t + 12\).

Step 5: Calculate the 100th Term

Substitute \(n = 100\) in the n-th term formula: \(a_{100} = -t + 3(100) - 3 = -t + 300 - 3 = -t + 297\). Therefore, the 100th term is \(-t + 297\).

Key Concepts

Understanding Common Difference

In the realm of arithmetic sequences, the term 'common difference' stands out as a fundamental concept. The common difference is the consistent gap between consecutive terms in an arithmetic sequence. Once you identify this difference, it becomes easier to predict future terms.

**Finding the Common Difference** To determine the common difference, simply subtract any term from the term that directly follows it. In the provided sequence, \(-t, -t+3, -t+6, -t+9, \dots\), you can find the common difference by calculating \((-t+3) - (-t) = 3\).

This difference remains the same for all pairs of consecutive terms in the sequence, hence why it's called 'common'. Recognizing the structure of an arithmetic sequence by identifying this value simplifies the process of working with these sequences.

The Power of the n-th Term Formula

The n-th term formula is pivotal when dealing with arithmetic sequences. This formula provides a way to find any term in the sequence without having to painstakingly list all the preceding terms.

**Breaking Down the Formula**

• For an arithmetic sequence, the n-th term, denoted as \(a_n\), can be calculated using the formula: \(a_n = a_1 + (n-1)d\). \
• Here, \a_1\ is the first term of the sequence, found as \a_1 = -t\ in our sequence.
• The value \(d\) is the common difference, and \(n\) represents the position of a term in the sequence.

Applying this to our exercise, by substituting \a_1 = -t\ and \d = 3\, we arrive at the formula: \(a_n = -t + (n-1) \times 3\), which simplifies to \(a_n = -t + 3n - 3\).

With this formula, you can efficiently calculate terms at any position like the fifth or 100th without listing each term individually.

Diving into Arithmetic Progression

Arithmetic progression is a sequence of numbers in which the difference of any two successive members is a constant, known as the common difference. This pattern creates a harmonized sequence where each term grows by a steady amount.

**Recognizing an Arithmetic Progression**

• Every arithmetic sequence forms a straight-line relationship when plotted, solidifying its identity as a linear progression.
• The sequence \(-t, -t+3, -t+6, -t+9, \dots\) exemplifies this structure with its constant addition of 3.
• The beauty of arithmetic progression lies in its predictability, allowing us to derive subsequent terms using the common difference.

The concepts of arithmetic progression are essential not only for academic purposes but also in various real-world applications, such as calculating consistent daily savings or predicting long-term financial growth.