To prove that the sequence defined by the \( n \)th term \( u_n = 3n + 1 \) is arithmetic, we need to show that the difference between consecutive terms, known as the common difference, is constant.

### Step 1: Define the \( n \)th term
The \( n \)th term of the sequence is given by:
\[
u_n = 3n + 1
\]

### Step 2: Find the \( (n+1) \)th term
Substitute \( n+1 \) for \( n \) in the expression for \( u_n \):
\[
u_{n+1} = 3(n+1) + 1
\]
Simplify the expression:
\[
u_{n+1} = 3n + 3 + 1 = 3n + 4
\]

### Step 3: Calculate the common difference
The common difference \( d \) is found by subtracting \( u_n \) from \( u_{n+1} \):
\[
d = u_{n+1} - u_n = (3n + 4) - (3n + 1)
\]
Simplify the expression:
\[
d = 3n + 4 - 3n - 1 = 3
\]

### Step 4: Conclusion
Since the common difference \( d \) is constant (equal to 3), the sequence defined by \( u_n = 3n + 1 \) is indeed an arithmetic sequence.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How do you determine if a sequence is geometric?
2. What is the general form of an arithmetic sequence?
3. How can you find the sum of an arithmetic sequence?
4. What is the relationship between the common difference and the first term in an arithmetic sequence?
5. How do arithmetic and geometric sequences differ in their growth patterns?

**Tip:** To quickly check if a sequence is arithmetic, compute the difference between consecutive terms. If the difference is constant, the sequence is arithmetic.

Learn how to prove that the sequence defined by u_n = 3n + 1 is arithmetic. Understand the concept of arithmetic sequences and the steps involved in verifying their properties. Suitable for students in Grades 7-9.

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