Problem 8 - Algebra and Trigonometry

Question

Write the first six terms of each arithmetic sequence. $$ a_{1}=\frac{3}{4}, d=-\frac{1}{4} $$

Step-by-Step Solution

Answer

The first six terms of the arithmetic sequence are \(\frac{3}{4}, \frac{1}{2}, \frac{1}{4}, 0, -\frac{1}{4}, -\frac{1}{2}\)

Step 1: Identify the given values

Identify the given values in the problem. Here, the first term of the arithmetic sequence \(a_{1}\) is \(\frac{3}{4}\) and the common difference \(d\) is \(-\frac{1}{4}\).

Step 2: Calculate the second term

Use the formula of the nth term of an arithmetic sequence to find the second term. Substitute \(n = 2\), \(a_{1} = \frac{3}{4}\), and \(d = -\frac{1}{4}\) into the formula: \(a_2 = a_1 + (2-1) \cdot d = \frac{3}{4} - \frac{1}{4} = \frac{1}{2}\). So, the second term is \(\frac{1}{2}\) .

Step 3: Calculate the third term

Repeat the same for the third term. Substitute \(n = 3\), \(a_{1} = \frac{3}{4}\), and \(d = -\frac{1}{4}\) into the formula: \(a_3 = a_1 + (3-1) \cdot d = \frac{3}{4} - 2 \cdot \frac{1}{4} = \frac{1}{4}\). So, the third term is \(\frac{1}{4}\) .

Step 4: Calculate the fourth term

Repeat the same for the fourth term. Substitute \(n = 4\), \(a_{1} = \frac{3}{4}\), and \(d = -\frac{1}{4}\) into the formula: \(a_4 = a_1 + (4-1) \cdot d = \frac{3}{4} - 3 \cdot \frac{1}{4} = 0\). So, the fourth term is 0 .

Step 5: Calculate the fifth term

Repeat the same for the fifth term. Substitute \(n = 5\), \(a_{1} = \frac{3}{4}\), and \(d = -\frac{1}{4}\) into the formula: \(a_5 = a_1 + (5-1) \cdot d = \frac{3}{4} - 4 \cdot \frac{1}{4} = -\frac{1}{4}\). So, the fifth term is -\frac{1}{4} .

Step 6: Calculate the sixth term

Repeat the same for the sixth term. Substitute \(n = 6\), \(a_{1} = \frac{3}{4}\), and \(d = -\frac{1}{4}\) into the formula: \(a_6 = a_1 + (6-1) \cdot d = \frac{3}{4} - 5 \cdot \frac{1}{4} = -\frac{1}{2}\). So, the sixth term is -\frac{1}{2} .

Key Concepts

Arithmetic Sequence Formula

An arithmetic sequence is a list of numbers with a specific pattern where each term after the first is found by adding a constant, known as the common difference, to the previous term. The arithmetic sequence formula, crucial for understanding arithmetic progression, is given by:

\[ a_n = a_1 + (n - 1)d \]
In this formula,

• \( a_n \) represents the nth term of the sequence,
• \( a_1 \) is the first term,
• \( n \) is the term number, and
• \( d \) is the common difference between terms.

The sequence's behavior is heavily dependent on the common difference. If \( d \) is positive, the sequence increases; if negative, it decreases. Understanding how to apply this formula is essential for efficiently solving problems related to arithmetic sequences.

Finding the Nth Term

Finding the nth term of an arithmetic sequence is a fundamental skill in sequence analysis. The task is straightforward once you've identified the first term and the common difference. By substituting these values into the arithmetic sequence formula, you can determine any term in the sequence.

For example, to find the sixth term given the first term \( a_1 = \frac{3}{4} \) and the common difference \( d = -\frac{1}{4} \), we substitute into the formula: \( a_6 = \frac{3}{4} + (6-1)(-\frac{1}{4}) \). This essential technique enables students to navigate through arithmetic progressions regardless of the sequence's length or complexity.

Arithmetic Progression

An arithmetic progression is another term for an arithmetic sequence. It refers to an ordered set of numbers in which the difference between consecutive terms is always the same. This constant difference is a critical feature of the progression and is what defines it as arithmetic.

Arithmetic progressions are frequently used in various mathematical contexts, from simple counting to more complex financial calculations. Recognizing an arithmetic progression allows us to predict future terms and sum large series, making it a highly valuable concept in both pure and applied mathematics.

Sequence and Series Arithmetic

Sequence and series arithmetic deals with the analysis of ordered lists where terms follow a specific rule, and the summing of these terms, respectively. While a sequence refers to the list of numbers itself, a series is about the sum of those numbers.

A key concept when dealing with series, especially arithmetic series, is the sum formula, which allows for the efficient calculation of the sum of all terms within the sequence. The ability to distinguish between sequences and series, and the application of their respective arithmetic formulas, is a staple in understanding a wider range of mathematical problems, as well as in everyday problem-solving scenarios.