Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. Given the \(n\) th term and the common difference of an arithmetic sequence, it is possible to find the \((n+1)\) th term.

Step-by-Step Solution

Answer

The given statement is True.

Step 1: Understanding the Problem

We are given the nth term and the common difference of an arithmetic sequence. We need to determine if we can calculate the \((n+1)\)th term from this data. Knowing the basic properties of an arithmetic sequence, it can be inferred that the \((n+1)\)th term can be gotten by adding the common difference to the nth term.

Step 2: Checking the Statement

Let's consider the general form of the nth term of an arithmetic sequence: \(a_n = a_1 + (n-1)d\). If we apply this formula to find the \((n+1)\)th term, we have: \(a_{n+1} = a_1 + (n)d\). However, we know that \(a_n = a_1 + (n-1)d\), hence: \(a_{n+1}= a_n + d\). This equation confirms that we can find the next term of an arithmetic sequence if we know the nth term and the common difference.

Step 3: Concluding

So, from the analysis, we can conclude that the given statement is true. You can indeed find the \((n+1)\)th term of an arithmetic sequence if you know the nth term and the common difference.

Key Concepts

Common Difference

The common difference in an arithmetic sequence is a key concept. It signifies the constant amount added to each term to get the next term. Whenever a sequence follows a specific pattern with this constant difference, it is identified as arithmetic. For example, in the sequence 2, 5, 8, 11, the common difference is 3.

To understand how the common difference is applied, consider an arithmetic sequence formula. The common difference, often symbolized as \(d\), is used as follows:

• Calculate each subsequent term by adding \(d\) to the current term.
• This keeps the sequence consistent and makes calculations predictable.

The concept is fundamental not only for identifying arithmetic sequences but also for continuing the sequence correctly.

nth Term

The \(n\)th term of an arithmetic sequence represents any term in the sequence that you want to determine. This term is crucial for finding other related terms or analyzing the sequence's properties. The general formula for the \(n\)th term is:
\[a_n = a_1 + (n-1)d\]
Here,

• \(a_n\) is the \(n\)th term.
• \(a_1\) stands for the first term of the sequence.
• \(d\) is the common difference.

Using this formula, you can calculate the value of any term in the sequence once you know the first term and the common difference.

This allows for flexibility and offers a straightforward method to find sequence elements.

(n+1)th Term

Determining the \((n+1)\)th term in an arithmetic sequence is an extension of understanding the \(n\)th term. Once you have the \(n\)th term, finding the next term is simple. You just add the common difference \(d\) to the \(n\)th term.

Let's use the expression: \(a_{n+1} = a_n + d\), which signifies:

• The next term \(a_{n+1}\) is attained by simply adding \(d\) to the \(n\)th term \(a_n\).
• This step-by-step process ensures that the sequence stays arithmetic and follows the defined pattern reliably.

The ability to find the \((n+1)\)th term ensures that you can continue the sequence seamlessly, given any starting point within the sequence.