To determine the $n$th term of the sequence $1, 8, 15, 22, 29, \ldots$, we need to analyze the pattern in the sequence.

Step 1: Identify the Common Difference

Let's look at the differences between consecutive terms:

• $8 - 1 = 7$
• $15 - 8 = 7$
• $22 - 15 = 7$
• $29 - 22 = 7$

The sequence has a common difference of 7, meaning it is an arithmetic sequence.

Step 2: Use the Formula for the $n$th Term of an Arithmetic Sequence

The general formula for the $n$th term of an arithmetic sequence is: $a_n = a_1 + (n - 1) \cdot d$ where:

• $a_1$ is the first term of the sequence,
• $d$ is the common difference,
• $n$ is the term number.

In this case:

• $a_1 = 1$,
• $d = 7$.

Step 3: Substitute the Values

Substituting these values into the formula gives: $a_n = 1 + (n - 1) \cdot 7$ $a_n = 1 + 7n - 7$ $a_n = 7n - 6$

Conclusion

The $n$th term of the sequence is: $a_n = 7n - 6$

Would you like further explanation or have any other questions?

Here are some related questions:

1. How do you determine the common difference in an arithmetic sequence?
2. What is the formula for the sum of the first $n$ terms in an arithmetic sequence?
3. How does the formula for the $n$th term change if the common difference is negative?
4. Can you determine the $n$th term if the sequence is not arithmetic?
5. What happens to the $n$th term formula if the first term is zero?

Tip: Always check if a sequence is arithmetic or geometric by analyzing the differences between terms.