Problem 15 - Beginning and Intermediate Algebra

Question

Write the first five terms of each arithmetic sequence with the given first term and common difference. $$a_{1}=-10, d=3$$

Step-by-Step Solution

Answer

The first five terms of the arithmetic sequence are: \(a_1 = -10,\ a_2 = -7,\ a_3 = -4,\ a_4 = -1,\ a_5 = 2\).

Step 1: Write down the given information

We have: \(a_1 = -10\) and common difference \(d = 3\).

Step 2: Find the second term of the sequence

To find the second term, we add the common difference to the first term: \[a_2 = a_1 + d = -10 + 3 = -7\]

Step 3: Find the third term of the sequence

To find the third term, we add the common difference to the second term: \[a_3 = a_2 + d = -7 + 3 = -4\]

Step 4: Find the fourth term of the sequence

To find the fourth term, we add the common difference to the third term: \[a_4 = a_3 + d = -4 + 3 = -1\]

Step 5: Find the fifth term of the sequence

To find the fifth term, we add the common difference to the fourth term: \[a_5 = a_4 + d = -1 + 3 = 2\]

Step 6: Write down the first five terms of the sequence

The first five terms of the sequence are: \[a_1 = -10,\ a_2 = -7,\ a_3 = -4,\ a_4 = -1,\ a_5 = 2\]

Key Concepts

Common Difference

The common difference in an arithmetic sequence is the amount that each term increases or decreases from the previous term. It is a crucial part of understanding arithmetic sequences, as it defines the sequence's pattern.
In the exercise provided, the common difference \(d\) is given as \(3\). This means that you add \(3\) to each term to get the next term. Here's a simple breakdown:

• Start at \(-10\)
• Add \(3\) to get \(-7\)
• Continue adding \(3\) each time

Recognizing this pattern helps us predict any term in the sequence without listing all previous ones, by using the formula \(a_n = a_1 + (n - 1) \cdot d\), where \(n\) is the term number. This formula is handy for quickly calculating any term once you've got the common difference.

Terms of a Sequence

In any sequence, each number is called a term. In an arithmetic sequence, terms are crucial elements that follow a specific order.
In the original exercise, you were tasked with finding the first five terms of an arithmetic sequence given the starting point \(a_1 = -10\) and the common difference \(d = 3\). The process involved was step-by-step:

• First term: \(a_1 = -10\)
• Second term: \(a_2 = -7\)
• Third term: \(a_3 = -4\)
• Fourth term: \(a_4 = -1\)
• Fifth term: \(a_5 = 2\)

By understanding each term's relationship with the others and the common difference, you create a predictable sequence. Observing the pattern also means you don’t need to start from scratch each time—you can just pick up from any term you know.

Algebraic Expressions

Algebraic expressions offer a way to express the rules that govern arithmetic sequences.
Typically, the general formula for the \(n\)-th term of an arithmetic sequence is given by the expression:

• \(a_n = a_1 + (n - 1) \cdot d\)

This expression is powerful because it lets you calculate any term without having to write down the whole sequence. In the exercise, the formula illustrates how to find any term by plugging in the known common difference and first term. For example, if you wanted to find the third term, you would set \(n = 3\) and compute:

• \(a_3 = -10 + (3-1) \times 3 = -4\)

Practicing with algebraic expressions enables quick computation and a deeper understanding of how sequences function.