Problem 18 - Algebra 2

Question

Find the 32nd term of each sequence. \(23,30,37,44, \dots\)

Step-by-Step Solution

Answer

The 32nd term of the sequence is 240

Step 1: Identify the sequence characteristics

The first term \(a_1\) of the sequence is 23 and the common difference \(d\) between each consecutive term is 7.

Step 2: Substitute values in the formula

now substitute the known values into the arithmetic sequence formula \(a_n = a_1 + (n-1)d\). This gives \(a_{32} = 23 + (32 - 1) \times 7 \)

Step 3: Calculate the 32nd term

Solving the equation gives: \(a_{32} = 23 + 31 \times 7 = 23 + 217 = 240\)

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, the common difference is crucial. It's the consistent amount you add to each term to get to the next one. To find the common difference, simply subtract the first term from the second term. For example, in the sequence provided (23, 30, 37, 44, ...), you can calculate the common difference as follows:

• Subtract 23 from 30, which gives 7.
• Check the next pair to verify: Subtract 30 from 37, which again gives 7.

This tells us that the common difference is 7. Understanding this value is key to predicting future terms in the sequence, as it remains constant throughout.

Exploring the Nth Term Formula

The nth term formula allows us to find any term in the arithmetic sequence without listing all the preceding terms. This formula is expressed as:

\[a_n = a_1 + (n-1) \times d\]

In this formula:

• \(a_n\) represents the nth term we want to find.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference.
• \(n\) is the term number.

Substitute these values for the given sequence to find any term. For instance, to find the 32nd term:

\[a_{32} = 23 + (32-1) \times 7\]

Calculating gives us \(a_{32} = 240\). This indicates the power of the nth term formula in quickly determining the position of any term in the sequence.

Characteristics of an Arithmetic Sequence

Arithmetic sequences have distinctive characteristics that differentiate them from other types of sequences. Let's explore these features:

• Constant Difference: As discussed, the difference between consecutive terms is always the same - this is known as the common difference. It gives the sequence a linear growth pattern.
• Predictability: Due to the constant difference, predicting future terms becomes straightforward. This predictability is one of the main advantages of arithmetic sequences.
• Graph Representation: If you graph an arithmetic sequence with the term number on the x-axis and the term value on the y-axis, it will form a straight line. This linear graph is due to the constant incremental change.

Understanding these characteristics can help you easily identify and work with arithmetic sequences in a variety of mathematical problems.