To find the number of terms $n$ in the arithmetic sequence, we start by recognizing that the sequence given is an arithmetic progression (AP).

Given:

• The first term $a_1 = 9$
• The common difference $d$ is the difference between any two consecutive terms: $d = 17 - 9 = 8$
• The sum of the sequence is $S_n = 636$

Formula for the sum of the first $n$ terms of an AP:

$S_n = \frac{n}{2} \cdot (2a_1 + (n-1) \cdot d)$ Substituting the given values into the formula:

$636 = \frac{n}{2} \cdot \left(2 \cdot 9 + (n-1) \cdot 8\right)$

Simplifying further:

$636 = \frac{n}{2} \cdot \left(18 + 8n - 8\right)$

$636 = \frac{n}{2} \cdot (8n + 10)$

Multiplying both sides by 2 to eliminate the fraction:

$1272 = n \cdot (8n + 10)$

Expanding and rearranging the equation:

$1272 = 8n^2 + 10n$

$8n^2 + 10n - 1272 = 0$

This is a quadratic equation in the form $an^2 + bn + c = 0$, where:

• $a = 8$
• $b = 10$
• $c = -1272$

Solving the quadratic equation using the quadratic formula:

$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Substituting the values:

$n = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 8 \cdot (-1272)}}{16}$

$n = \frac{-10 \pm \sqrt{100 + 40704}}{16}$

$n = \frac{-10 \pm \sqrt{40804}}{16}$

$n = \frac{-10 \pm 202}{16}$

This gives two possible values for $n$:

$n = \frac{192}{16} = 12$

or

n = \frac{-212}{16} \approx -13.25 \quad \text{(which is not possible since n must be positive)}

Thus, the number of terms $n$ in the arithmetic sequence is $\boxed{12}$.

Would you like to go deeper into any part of this solution?

Here are some related questions to consider:

1. What is the 12th term of this arithmetic sequence?
2. How would you find the sum of the first 20 terms of the same sequence?
3. What would the sum be if there were 10 terms instead of 12?
4. Can you derive a general formula for the sum of any arithmetic sequence?
5. How would the common difference change the sum if it were doubled?
6. What if the sequence started at a different number, say 5?
7. How does the sum change if we remove the last term from the sequence?
8. How would you graph the sequence and its cumulative sum?

Tip: When solving quadratic equations, always check both possible solutions to ensure that the one chosen is valid in the context of the problem.