Question

Show that each sequence is arithmetic. Find the common difference, and list the first four terms. $$ \left\\{b_{n}\right\\}=\\{3 n+1\\} $$

Short answer

The sequence is arithmetic with a common difference of 3 and the first four terms are 4, 7, 10, and 13.

Step-by-step solution

- Determine the General Form

The given sequence is defined by the general term \( b_n = 3n + 1 \).

- Calculate the First Term

To find the first term, substitute n = 1 into the general term: \( b_1 = 3(1) + 1 = 4 \).

- Calculate the Second Term

Substitute n = 2 into the general term to get the second term: \( b_2 = 3(2) + 1 = 7 \).

- Calculate the Third Term

Substitute n = 3 into the general term to get the third term: \( b_3 = 3(3) + 1 = 10 \).

- Calculate the Fourth Term

Substitute n = 4 into the general term to get the fourth term: \( b_4 = 3(4) + 1 = 13 \).

- Verify the Common Difference

To check if the sequence is arithmetic, find the difference between consecutive terms: \( b_2 - b_1 = 7 - 4 = 3 \), \( b_3 - b_2 = 10 - 7 = 3 \), and \( b_4 - b_3 = 13 - 10 = 3 \). The common difference d is 3, confirming that the sequence is arithmetic.

- Summarize the Findings

The sequence \( \{b_n\} = 3n + 1 \) is arithmetic with a common difference of 3. The first four terms are 4, 7, 10, and 13.

Key concepts

Common Difference

In an arithmetic sequence, the 'common difference' is a key element that separates an arithmetic sequence from other types of sequences. It is the constant difference between consecutive terms in the sequence.
Finding the common difference involves a simple subtraction:
you subtract the first term from the second term, the second term from the third, and so forth.

For our sequence defined by \( b_n = 3n + 1 \), we can calculate the common difference by subtracting:

• \( b_2 - b_1 = 7 - 4 = 3 \)
• \( b_3 - b_2 = 10 - 7 = 3 \)
• \( b_4 - b_3 = 13 - 10 = 3 \)

In each case, the difference is 3.
This means our common difference, denoted as \(d\), is 3.
The value of \(d\) is crucial in proving whether a sequence is arithmetic. If the difference remains constant throughout, the sequence is indeed an arithmetic sequence.

General Term

The general term of an arithmetic sequence provides a formula to find any term in the sequence. It is often expressed in the form
\[ b_n = a + (n-1)d \]
where:

• \(a\) is the first term
• \(d\) is the common difference
• \(n\) is the term number (or position in the sequence)

In our specific exercise, the sequence is defined by the general term:
\( b_n = 3n + 1 \)
Here, compare it with the standard form, we identify:
\( a = 4 \) (calculated as the first term when \( n = 1 \))
\( d = 3 \) (our common difference).
Having a general term allows us to find the value of any term in the sequence without having to manually calculate each previous term. For instance, if we need to find the 10th term, we just substitute \( n = 10 \) into the general term formula:
\[ b_{10} = 3*10 + 1 = 30 + 1 = 31 \]

Sequence Verification

Verifying whether a sequence is arithmetic involves checking the common difference throughout the sequence. If the common difference is constant, the sequence is arithmetic.
In our exercise, we already calculated the first few terms using:
\( b_1 = 3(1) + 1 = 4 \)
\( b_2 = 3(2) + 1 = 7 \)
\( b_3 = 3(3) + 1 = 10 \)
\( b_4 = 3(4) + 1 = 13 \)
Next, we checked the differences between consecutive terms:

• \( b_2 - b_1 = 7 - 4 = 3 \)
• \( b_3 - b_2 = 10 - 7 = 3 \)
• \( b_4 - b_3 = 13 - 10 = 3 \)

Since the difference is consistently 3, we conclude that the sequence is arithmetic with a common difference of 3.

First Four Terms

Listing the first four terms of an arithmetic sequence helps in verifying its properties and better understanding its structure.
Given the sequence defined by \( b_n = 3n + 1 \), we use this formula to get the first four terms by substituting \( n\) values from 1 to 4:

• First term: \( b_1 = 3(1) + 1 = 4 \)
• Second term: \( b_2 = 3(2) + 1 = 7 \)
• Third term: \( b_3 = 3(3) + 1 = 10 \)
• Fourth term: \( b_4 = 3(4) + 1 = 13 \)

These steps give us the first four terms: 4, 7, 10, and 13. By examining these terms, we can visibly see the common difference of 3 between each consecutive term, supporting our understanding and verification of the sequence.