Problem 4 - Algebra and Trigonometry

Question

Write the first six terms of each arithmetic sequence. $$ a_{1}=-8, d=5 $$

Step-by-Step Solution

Answer

The first six terms of the sequence are -8, -3, 2, 7, 12, 17

Step 1: Understand the Problem

The problem provides the first term \(a_{1}\) of an arithmetic sequence as -8 and the common difference \(d\) as 5. The task is to find the first six terms of this sequence.

Step 2: Find the Second Term

The second term \(a_{2}\) of an arithmetic sequence is found by adding the common difference to the first term, i.e., \(a_{2} = a_{1} + d = -8 + 5 = -3\)

Step 3: Find the Third Term

The third term \(a_{3}\) of an arithmetic sequence is found by adding the common difference to the second term, i.e., \(a_{3} = a_{2} + d = -3 + 5 = 2\)

Step 4: Find the Fourth Term

The fourth term \(a_{4}\) of an arithmetic sequence is found by adding the common difference to the third term, i.e., \(a_{4} = a_{3} + d = 2 + 5 = 7\)

Step 5: Find the Fifth Term

The fifth term \(a_{5}\) of an arithmetic sequence is found by adding the common difference to the fourth term, i.e., \(a_{5} = a_{4} + d = 7 + 5 = 12\)

Step 6: Find the Sixth Term

The sixth term \(a_{6}\) of an arithmetic sequence is found by adding the common difference to the fifth term, i.e., \(a_{6} = a_{5} + d = 12 + 5 = 17\)

Key Concepts

Common Difference

In an arithmetic sequence, the term "common difference" refers to the consistent amount that is added (or subtracted) from one term to the next. This value is denoted by the letter \(d\). It's called "common" because it remains the same throughout the entire sequence. Understanding the common difference is crucial because it determines how the sequence progresses from one term to another.

• If the common difference is positive, the sequence will increase gradually.
• If the common difference is negative, the sequence will decrease.

In our exercise, the common difference is 5. This means every term in the sequence is 5 units greater than the previous term. Recognizing this pattern helps predict any term in the sequence.

First Term

The first term in an arithmetic sequence, signified by \(a_1\), is the starting point of the sequence. It's the very first element and forms the foundation of the series. Knowing the first term is essential because it establishes the base value from which all other terms are derived.

In the provided exercise, the first term is \(-8\). This simple piece of information sets the stage for finding the subsequent terms using the common difference. Without a first term, we wouldn't have a point to begin calculations.

Terms of a Sequence

The terms of a sequence are simply the individual elements or numbers that make up the sequence. In an arithmetic sequence, each term follows the previous one by adding the common difference.

• Starting from the first term, each subsequent term can be found by adding the common difference.
• This creates a straightforward pattern that continues infinitely, unless specified otherwise.

For instance, if the first term is \(-8\) and the common difference is 5, then the terms in the sequence would be \(-8, -3, 2, 7, 12, 17, ... \)continuing in this manner indefinitely. Understanding the terms and how they're derived allows for predicting any term within the sequence.

Finding Terms

To find any specific term in an arithmetic sequence, we can use the formula:\[a_n = a_1 + (n - 1) \, d\]where \(a_n\) is the term you're trying to find, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

The exercise illustrates this well. Given that \(a_1 = -8\) and the common difference \(d = 5\), the process involves simply plugging these values into the formula to find terms like \(a_2, a_3,\) and so forth.

• For \(a_2\), we get \(-8 + 5 = -3\).
• For \(a_3\), it's \(-3 + 5 = 2\).
• This pattern continues as we find \(a_4, a_5,\) and \(a_6\).

This systematic approach ensures that you can quickly and accurately determine any term in a sequence, just by knowing two key pieces of information: the first term and the common difference.