This is a problem about finding the number of terms in an arithmetic sequence. Here's how to solve it step by step:

1. Arithmetic Sequence Formula: The $n$-th term ($a_n$) of an arithmetic sequence is given by: $a_n = a_1 + (n-1)d$ where:

   • $a_1 = 12$ (first term),
   • $d = 8$ (common difference),
   • $a_n = 3596$ (final value to reach).

2. Set up the equation: Substitute the values into the formula: $3596 = 12 + (n-1) \cdot 8$

3. Simplify: $3596 - 12 = (n-1) \cdot 8$ $3584 = (n-1) \cdot 8$

4. Solve for $n-1$: Divide both sides by 8: $n-1 = \frac{3584}{8} = 448$

5. Find $n$: Add 1: $n = 448 + 1 = 449$

Final Answer:

The number of terms is 449.

Would you like a detailed explanation of any step?

Related Questions:

1. What if the common difference was negative? How would the sequence behave?
2. How is the sum of an arithmetic sequence calculated for $n$ terms?
3. Can you derive the general formula for $n$ without substituting values?
4. What is the formula for the $k$-th term in terms of $k$ instead of $n$?
5. How does this sequence graphically appear on a number line?

Tip:

Always double-check your arithmetic operations when solving equations involving sequences!