Let's solve the problem step-by-step by finding the nth-term expressions for the given sequences.

a) Sequence: $1, 4, 7, 10, 13$

This sequence has a constant difference of 3 between each term. It is an arithmetic sequence.

To find the nth term ($T_n$) of an arithmetic sequence, we use the general formula: $T_n = a + (n - 1)d$ Where:

• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

For this sequence:

• $a = 1$ (first term),
• $d = 3$ (common difference).

Substituting into the formula: $T_n = 1 + (n - 1) \cdot 3$ Simplifying: $T_n = 1 + 3n - 3 = 3n - 2$

Thus, the nth term for sequence (a) is: $T_n = 3n - 2$

b) Sequence: $8, 10, 12, 14, 16$

This sequence has a constant difference of 2 between each term, so it is also an arithmetic sequence.

Using the same formula for the nth term: $T_n = a + (n - 1)d$ Where:

• $a = 8$ (first term),
• $d = 2$ (common difference).

Substituting into the formula: $T_n = 8 + (n - 1) \cdot 2$ Simplifying: $T_n = 8 + 2n - 2 = 2n + 6$

Thus, the nth term for sequence (b) is: $T_n = 2n + 6$

Final Answers:

• Sequence (a): $T_n = 3n - 2$
• Sequence (b): $T_n = 2n + 6$

Would you like more details, or have any questions about this solution?

Here are 5 follow-up questions related to this:

1. How do you derive the nth term for a geometric sequence?
2. Can you find the sum of the first $n$ terms for both sequences?
3. What would the 10th term be for both sequences?
4. How would the nth term formula change if the common difference was negative?
5. Can you create a recursive formula for these sequences?

Tip: When dealing with arithmetic sequences, always check the common difference to ensure the pattern holds before applying formulas.