Let’s solve this arithmetic sequence problem.

Given:

• First three terms of the sequence: $7, 11, 15, \dots$
• Formula for the $n$-th term of an arithmetic sequence:
  $a_n = a_1 + (n-1)d$ where:
  • $a_1$ is the first term,
  • $d$ is the common difference,
  • $n$ is the term number.

Step 1: Find the common difference ($d$)

The common difference $d$ is the difference between consecutive terms: $d = 11 - 7 = 4$

Step 2: Find $a_8$

Using the formula for $a_n$: $a_8 = a_1 + (8-1)d = 7 + 7 \cdot 4 = 7 + 28 = 35$

So, $a_8 = 35$.

Step 3: General formula for $a_n$

Substitute $a_1 = 7$ and $d = 4$ into the general formula: $a_n = 7 + (n-1)4 = 7 + 4n - 4 = 4n + 3$

Thus, the formula for the $n$-th term is: $a_n = 4n + 3$

Final Answers:

1. $a_8 = 35$
2. $a_n = 4n + 3$

Would you like further clarification or help with other examples?

Related Questions:

1. How do we derive the general formula for an arithmetic sequence step by step?
2. What is the 15th term ($a_{15}$) of this sequence?
3. Can the sum of the first 8 terms ($S_8$) of this sequence be calculated?
4. What would $n$ be if $a_n = 99$?
5. How do arithmetic sequences differ from geometric sequences in their formulas?

Tip:

When working with arithmetic sequences, always calculate the common difference first—it’s key to solving all related problems!