Question

Find the 23 rd term of an arithmetic sequence with first term 2 and common difference \(7 .\)

Short answer

Answer: The 23rd term of the arithmetic sequence is 156.

Step-by-step solution

Understand the arithmetic sequence formula

The arithmetic sequence formula is used to find the nth term of an arithmetic sequence. It is given by: \(T_n = a + (n-1)d\) For our given problem: \(a = 2\): first term \(n = 23\): term position \(d = 7\): common difference

Plug values into the formula

Now that we have all the required values, we can plug them into the arithmetic sequence formula: \(T_n = a + (n-1)d\) And place all known values: \(T_{23} = 2 + (23-1)7\)

Simplify the expression

Now we'll simplify the expression: \(T_{23} = 2 + (22)7\) \(T_{23} = 2 + 154\) \(T_{23} = 156\)

State the final answer

The 23rd term of the arithmetic sequence with first term 2 and common difference 7 is: \(T_{23} = 156\)

Key concepts

nth term formula

In arithmetic sequences, finding a specific term is straightforward thanks to the nth term formula. This powerful tool allows us to calculate any term in an arithmetic sequence without listing out all the preceding terms. Think of it as your shortcut to efficiency!

The nth term formula is expressed as:\[ T_n = a + (n-1)d \]where:

• \(T_n\): represents the nth term you want to find.
• \(a\): is the first term of the sequence.
• \(n\): is the position of the term.
• \(d\): is the common difference between consecutive terms.

By using this formula, you simply plug the given values into it, and perform basic arithmetic to find your desired term. This saves time and prevents mistakes compared to adding the common difference repeatedly. Whether the sequence is short or long, the nth term formula keeps your math quick and easy.

common difference

The common difference is a key characteristic of any arithmetic sequence. Simply put, it is the consistent amount added or subtracted from one term to arrive at the next.

In our original exercise, the common difference is \(7\). This means that every term in the sequence is 7 units greater than the term before. The beauty of the common difference lies in its simplicity: its value remains unchanged throughout the sequence.

To find the common difference:

• Subtract any term from the subsequent term in the sequence.
• Check the result remains constant between any two consecutive terms.

Understanding the concept of the common difference helps unravel the pattern of the sequence and sets the foundation for using the nth term formula efficiently. It's the backbone that keeps the arithmetic sequence structured and predictable.

sequence problem-solving

Solving sequence problems, especially arithmetic ones, involves a systematic approach. Here’s how you can tackle them efficiently:

Identification: Begin by identifying whether the sequence is arithmetic. This involves spotting a common difference between the terms. If each term maintains a consistent gap, you’re dealing with an arithmetic sequence.
Applying the nth term formula: Once confirmed, the next step is using the nth term formula. By substituting the appropriate values for the first term, common difference, and the term position, you can directly calculate the desired term without listing intermediate terms.
Checking your work: Always re-evaluate your calculations by simplifying carefully. Recalculate steps if needed to ensure accuracy.

Approaching sequence problem-solving with these strategies reduces errors and boosts confidence. This methodical process is crucial for quantitative reasoning and enhances overall problem-solving skills.