Problem 29 - Beginning and Intermediate Algebra

Question

For each arithmetic sequence, find \(a_{n}\) and then use \(a_{n}\) to find the indicated term. $$a_{1}=0, d=-5 ; a_{23}$$

Step-by-Step Solution

Answer

We are given the first term \(a_1 = 0\), the common difference \(d = -5\), and need to find the 23rd term (\(a_{23}\)). The general formula for the nth term of an arithmetic sequence is \(a_n = a_1 + (n-1)d\). To find the 23rd term, use the given values and \(n=23\): \[a_{23} = 0 + (22)(-5)\] \[a_{23} = -110\] The 23rd term, \(a_{23}\), is equal to \(-110\).

Step 1: Write down the given information

We are given the first term \(a_1 = 0\), the common difference \(d = -5\), and are asked to find the 23rd term \(a_{23}\).

Step 2: Find the general formula for the nth term

The general formula for the nth term of an arithmetic sequence is \(a_n = a_1 + (n-1)d\). This formula allows us to find any term in the sequence if we know the first term, the common difference, and the position of the term we're trying to find.

Step 3: Calculate the 23rd term

To find the 23rd term, we will use the formula for \(a_n\) with the given values and \(n=23\). So, we have: \[a_{23} = a_1 + (23-1)d\] \[a_{23} = 0 + (22)(-5)\]

Step 4: Solve for the 23rd term

Now, solve the equation: \[a_{23} = 0 + (22)(-5)\] \[a_{23} = 0 - 110\] \[a_{23} = -110\] The 23rd term, \(a_{23}\), is equal to \(-110\).

Key Concepts

nth term formula

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. To find any specific term in such a sequence, we use the **nth term formula**. The formula is expressed as:\[a_n = a_1 + (n-1) \cdot d\]
The components of this formula are:

• \(a_n\): The term you want to find.
• \(a_1\): The first term of the sequence.
• \(n\): The position of the term.
• \(d\): The common difference between terms.

The formula helps calculate any term quickly if the first term and the common difference are known. In our original exercise, using this formula allowed us to find the 23rd term based on the given first term and common difference.

common difference

Understanding the **common difference** is crucial in arithmetic sequences. The common difference, represented by \(d\), is the amount we add or subtract to get from one term to the next. It can be calculated as: \[d = a_{n+1} - a_n\]
In our example:

• First term \(a_1 = 0\)
• Common difference \(d = -5\)

Hence, the terms decrease by 5 for each subsequent element in the sequence. Recognizing this pattern allows easy computation of terms without finding all the previous ones. This difference simplifies predicting any term in the sequence, using the nth term formula.

sequence terms

The **sequence terms** of an arithmetic sequence are all the numbers that make up the sequence, each separated by the common difference. Each term in the sequence is related to the first term through consistent additions or subtractions of the common difference. In our example sequence starting with 0 and a common difference of -5, the first few terms would be:

• \(a_1 = 0\)
• \(a_2 = 0 - 5 = -5\)
• \(a_3 = -5 - 5 = -10\)

This arithmetic pattern continues indefinitely, following the rule established by the common difference. Comprehending sequence terms allows for easy visualization of how each term is formulated based on its position in the sequence.

mathematical sequences

**Mathematical sequences** involve ordered lists of numbers that follow particular rules. An arithmetic sequence, specifically, is characterized by adding a fixed number—the common difference—to obtain the next term. Other mathematical sequences could have different rules, like geometric sequences (where each term is multiplied by a constant ratio rather than being added). The sequence in our exercise only requires:

• A first term, \(a_1\)
• A common difference, \(d\)

This simplicity makes arithmetic sequences ideal for introducing the concept of sequences in mathematics. They provide a foundation for understanding more complex sequence structures in mathematical studies, serving as a gateway to exploring various sequence types and their unique properties.