Problem 3 - College Algebra

Question

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{1}=-7, d=4$$

Step-by-Step Solution

Answer

The first six terms of the arithmetic sequence are -7, -3, 1, 5, 9, 13.

Step 1: Identify the first term and common difference

It's given that the arithmetic sequence has the first term \(a_{1} = -7\) and the common difference \(d = 4\).

Step 2: Calculate the second term

Using the formula \(a_{2} = a_1 + d = -7 + 4 = -3\). So, the second term is -3.

Step 3: Calculate the third term

As per the formula, \(a_{3} = a_{2} + d = -3 + 4 = 1\). So, the third term is 1.

Step 4: Calculate the fourth term

Again, by the formula, \(a_{4} = a_{3} + d = 1 + 4 = 5\). The fourth term is 5.

Step 5: Calculate the fifth term

According to the formula, \(a_{5} = a_{4} + d = 5 + 4 = 9\). The fifth term is 9.

Step 6: Calculate the sixth term

Following the formula, \(a_{6} = a_{5} + d = 9 + 4 = 13\). So, the sixth term is 13.

Key Concepts

First Term

The first term of an arithmetic sequence is the starting point of the sequence and is crucial because it sets the stage for the entire pattern of numbers. In our example, the first term is given as \( a_{1} = -7 \). This is the number you start with and then build upon as you progress through the sequence. Knowing the first term allows you to begin the sequence and use it as a foundation for finding subsequent terms. Think of it like the first step you take on a journey.

Common Difference

An arithmetic sequence is characterized by a constant step called the common difference. This is the amount you add to each term to get the next one. In our example, this common difference is \( d = 4 \).

• The difference remains consistent throughout the sequence.
• It can be positive, increasing the sequence, or negative, decreasing it.
• In essence, it's the heartbeat of the sequence, dictating how it grows or shrinks.

The common difference enables us to calculate the next term in the sequence with ease.

Sequence Terms

Each term in the arithmetic sequence is derived by adding the common difference to the previous term, starting from the first term. Here’s how our sequence unfolds:

• The first term is \( -7 \).
• The second term is calculated as \( -7 + 4 = -3 \).
• The third term is \( -3 + 4 = 1 \).
• This continues with the series: \( 5, 9, 13 \) as fourth, fifth, and sixth terms.

These are called sequence terms, and they are part of the series you analyze or solve.

Mathematical Formula

The process of finding each term in the arithmetic sequence uses a straightforward mathematical formula: \[ a_{n} = a_{1} + (n-1) \cdot d \]This formula helps us calculate any term if we know the first term \( a_{1} \), the common difference \( d \), and the term's position \( n \).

• For the second term, apply: \( a_{2} = -7 + (2-1) \cdot 4 = -3 \).
• Similarly, to find the nth term, like the third or any other, plug into the formula.
• This mathematical approach is precise and saves time when calculating large sequences.

Using this formula ensures accuracy and ease when working with arithmetic sequences.