Problem 12 - College Algebra

Question

For the following exercises, write the first five terms of the arithmetic series given two terms. $$ a_{1}=17, a_{7}=-31 $$

Step-by-Step Solution

Answer

The first five terms are: 17, 9, 1, -7, -15.

Step 1: Determine the Common Difference

To find the common difference, we use the formula for the nth term of an arithmetic sequence: \( a_n = a_1 + (n-1) imes d \). Given that \( a_1 = 17 \) and \( a_7 = -31 \), substitute these values into the formula to set up the equation: \( -31 = 17 + 6d \). Solve for \( d \) by isolating it: \( -31 - 17 = 6d \), giving us \( -48 = 6d \). Therefore, \( d = -8 \).

Step 2: Calculate the First Five Terms

Now that we have the first term \( a_1 = 17 \) and the common difference \( d = -8 \), we can find the first five terms by repeatedly adding the common difference to the previous term: 1. \( a_1 = 17 \) 2. \( a_2 = a_1 + d = 17 - 8 = 9 \) 3. \( a_3 = a_2 + d = 9 - 8 = 1 \) 4. \( a_4 = a_3 + d = 1 - 8 = -7 \) 5. \( a_5 = a_4 + d = -7 - 8 = -15 \) Thus, the first five terms are 17, 9, 1, -7, and -15.

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, understanding the common difference is essential. The common difference, denoted as \( d \), is the consistent interval between each consecutive term. In other words, if you subtract any term from the following term, you'll find \( d \). This constant value is what makes an arithmetic sequence predictable.
For example, knowing that in the sequence given in the exercise \( a_1 = 17 \) and \( a_7 = -31 \), we used the nth term formula to find \( d \). We set up an equation \( -31 = 17 + 6d \). Solving this, we determined that the common difference is \( d = -8 \).
Understanding the common difference allows you to build the sequence term by term, ensuring that each term is simply the previous term added to \( d \). This creates a powerful tool to analyze and predict patterns in the sequence without needing all the terms laid out upfront.

Delving into Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each term after the first is formed by adding a fixed, constant amount to the previous term. This amount is known as the common difference. Arithmetic sequences are straightforward to navigate once you understand this pattern.

• The initial term (usually denoted \( a_1 \) ) starts your sequence. In the exercise, this was given as 17.
• Each term in the sequence is then the sum of the previous term and the common difference \( d \). As shown, when \( d = -8 \), it creates a descending sequence.
• The beauty of arithmetic sequences is in their predictability, which makes them easy to manage analytically, visually, or numerically.

When you're working with an arithmetic sequence, knowing just a single term and the common difference can help you construct the rest of the sequence, providing an effective way to generate terms and solve related problems.

Unpacking the Nth Term Formula

The nth term formula is a cornerstone of arithmetic sequences. This formula allows you to find any term in the sequence without having to individually calculate each preceding term. The formula is expressed as:
\[ a_n = a_1 + (n-1) \times d \]
This equation directly relates the desired term \( a_n \) to the first term \( a_1 \) and the position \( n \) of the term in the sequence using the common difference \( d \). Whether solving for a term towards the beginning or the end of a sequence, this formula provides a simple and straightforward calculation.

• In the exercise, the nth term formula helped us correctly determine \( a_7 \) as \( -31 \).
• The result from our calculations also solidified the common difference \( d = -8 \), which we then used to derive other aspects of the sequence.

This general formula is not just a tool for calculation; it reveals the elegant structure behind arithmetic sequences that allows clear insight into their developing pattern.