Question

In Problems 7 -16, show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{s_{n}\right\\}=\\{n+4\\} $$

Short answer

The sequence is arithmetic with a common difference of 1. The first four terms are 5, 6, 7, and 8.

Step-by-step solution

- Identify the general form of the sequence

The given sequence is defined by \(\{s_{n}\}=n+4\). This means for any term \(s_{n}\), the value is calculated by the formula \(n+4\).

- Calculate the common difference

For a sequence to be arithmetic, the difference between consecutive terms must be constant. Calculate the first term \(s_1\), the second term \(s_2\), and find the difference between them: \(s_1 = 1 + 4 = 5\) \(s_2 = 2 + 4 = 6\) The common difference, \(d\), is \(s_2 - s_1 = 6 - 5 = 1\).

- Verify the common difference with additional terms

To confirm the sequence is arithmetic, calculate the next two terms and check the differences. \(s_3 = 3 + 4 = 7\) \(s_4 = 4 + 4 = 8\) The differences are \(s_3 - s_2 = 7 - 6 = 1\) and \(s_4 - s_3 = 8 - 7 = 1\). The common difference is consistently \(1\).

- List the first four terms

The first four terms are \(s_1 = 5\), \(s_2 = 6\), \(s_3 = 7\), and \(s_4 = 8\).

Key concepts

Common Difference

The common difference is a key concept in understanding arithmetic sequences. In an arithmetic sequence, the difference between any two consecutive terms is always constant. This constant value is known as the common difference, often denoted by the letter \(d\).

In the given sequence \ \{s_{n}\} = n + 4 \, we start by identifying the first few terms to determine the common difference.

Let's calculate this step-by-step:

• 1st term: \(s_1 = 1 + 4 = 5\)
• 2nd term: \(s_2 = 2 + 4 = 6\)
• 3rd term: \(s_3 = 3 + 4 = 7\)
• 4th term: \(s_4 = 4 + 4 = 8\)

To find the common difference, we simply subtract each consecutive term:

\(s_2 - s_1 = 6 - 5 = 1\)
\(s_3 - s_2 = 7 - 6 = 1\)
\(s_4 - s_3 = 8 - 7 = 1\)

All differences are equal to 1, confirming that the common difference \(d\) is indeed 1. This consistent value tells us that the sequence is arithmetic.

Sequence Formula

The sequence formula is crucial for generating the terms of an arithmetic sequence. It provides a way to find any term in the sequence without listing all preceding terms. The general formula for the nth term, \(s_n\), in an arithmetic sequence is:

\(s_n = a + (n-1)d\)

Here, \(a\) is the first term, and \(d\) is the common difference.

However, in this specific exercise, we are given the sequence formula directly as:

\(s_n = n + 4\)

This formula simplifies the process, allowing us to find the value of any term by substituting \(n\) with the term's position number.

Let's verify it by finding the first four terms using the formula:

• For \(n = 1\), \(s_1 = 1 + 4 = 5\)
• For \(n = 2\), \(s_2 = 2 + 4 = 6\)
• For \(n = 3\), \(s_3 = 3 + 4 = 7\)
• For \(n = 4\), \(s_4 = 4 + 4 = 8\)

Each term aligns perfectly with our initial steps, confirming the formula's accuracy. This is a quick and reliable method to generate terms in the sequence.

First Terms

Listing the first terms of an arithmetic sequence helps to visually confirm the pattern and the common difference. The initial step in working with such sequences is often calculating these early terms.

Given the sequence formula \(s_n = n + 4\), let's find the first four terms to see the arithmetic progression unfold:

• 1st term, \(s_1\): Substituting \(n = 1\) gives us \(1 + 4 = 5\).
• 2nd term, \(s_2\): Substituting \(n = 2\) gives us \(2 + 4 = 6\).
• 3rd term, \(s_3\): Substituting \(n = 3\) gives us \(3 + 4 = 7\).
• 4th term, \(s_4\): Substituting \(n = 4\) gives us \(4 + 4 = 8\).

These calculations show the first four terms to be \(5, 6, 7, \text{ and } 8\). Listing these terms is a straightforward way to recognize the sequence's structure. It also aids in confirming that our common difference, \(d\), remains consistent throughout the sequence.

Furthermore, understanding these initial terms enhances comprehension and provides a foundation for solving more complex problems involving arithmetic sequences.