Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference. \(a_{1}=-25, d=-9\)

Step-by-Step Solution

Answer

The first five terms are -25, -34, -43, -52, and -61.

Step 1: Understand the Formula

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. This constant is known as the common difference (d). The general formula for an arithmetic sequence is \( a_n = a_1 + (n-1) \times d \), where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

Step 2: Calculate the Second Term

Using the formula \( a_n = a_1 + (n-1) \times d \), let's find the second term \( a_2 \). Substitute \( a_1 = -25 \), \( n = 2 \), and \( d = -9 \) into the formula: \( a_2 = -25 + (2-1) \times (-9) = -25 - 9 = -34 \).

Step 3: Calculate the Third Term

For the third term \( a_3 \), use the formula with \( n = 3 \): \( a_3 = -25 + (3-1) \times (-9) = -25 - 18 = -43 \).

Step 4: Calculate the Fourth Term

To find the fourth term \( a_4 \), use the formula with \( n = 4 \): \( a_4 = -25 + (4-1) \times (-9) = -25 - 27 = -52 \).

Step 5: Calculate the Fifth Term

Finally, use the formula to calculate the fifth term \( a_5 \) with \( n = 5 \): \( a_5 = -25 + (5-1) \times (-9) = -25 - 36 = -61 \).

Key Concepts

Understanding the Common Difference

The common difference, often denoted as \( d \), is a crucial part of arithmetic sequences. It is the fixed amount or interval by which each term in the sequence increases or decreases. In simpler terms, it tells us how much to add or subtract each time we move to the next term in the sequence.

In the given problem, the common difference is \(-9\). This means that each term in the sequence is \(-9\) less than the previous one. Understanding the common difference helps in predicting what the next few numbers in the sequence will be, without always having to perform complex calculations. By knowing this difference, you can easily move from one term to the next.

• For example, if the first term \( a_1 \) is \(-25\), then the second term \( a_2 \) can be calculated as \(-25 - 9 = -34\).
• Continuing this, you subtract \(9\) from \(-34\) to get the third term, \(-43\).

This continual subtraction pattern forms the essence of the sequence.

Using the Arithmetic Formula

The arithmetic formula is a powerful tool used to find any term in an arithmetic sequence. The formula is expressed as \( a_n = a_1 + (n-1) \times d \). Let’s break down the formula:

• \( a_n \): the term you want to find.
• \( a_1 \): the first term of the sequence.
• \( n \): the position of the term within the sequence.
• \( d \): the common difference, a constant added to each term from the start.

With this formula, you don't have to go step-by-step through each term. For example, to find the 5th term where \( a_1 = -25 \) and \( d = -9 \), substitute into the formula: \[a_5 = -25 + (5-1) \times (-9) = -25 - 36 = -61\]
The formula allows you to calculate any position in the sequence effectively, streamlining the process.

Exploring Sequence Terms

Sequence terms are individual numbers that make up a sequence. In an arithmetic sequence, each of these terms relates directly through the common difference. Each succeeding term is derived by adding the common difference to the previous term. This relation is what defines and structures the sequence.

In our example, the sequence starts with \(-25\) and continues with \(-34, -43, -52,\) and \(-61\). Each of these is calculated by using the arithmetic formula and adding the common difference repeatedly. Understanding sequence terms primarily involves recognizing that each term follows a predictable pattern governed by the common difference.

• The first term, \( -25 \), gives us our starting point.
• Successive terms like \(-34\), \(-43\), etc., follow due to the arithmetic formula, keeping a constant interval between them.

This predictable nature of arithmetic sequences can be very helpful in various mathematical problem-solving scenarios.