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For each of the summations, use the formula for the sum of the first n integers either to evaluate the sum or express it in closed form.

To solve the given summations, we can use the formula for the sum of an arithmetic series. The general formula for the sum of an arithmetic series is:

\[
S_n = \frac{n}{2} \times (a + l)
\]

where:
- \( S_n \) is the sum of the series,
- \( n \) is the number of terms,
- \( a \) is the first term, and
- \( l \) is the last term.

Alternatively, if you know the number of terms \( n \), the sum can also be expressed as:

\[
S_n = \frac{n}{2} \times (2a + (n-1) \times d)
\]

where \( d \) is the common difference between terms (which is 1 in this case).

### Part (a) - Sum from 9 to 600

The arithmetic series starts at 9 and ends at 600. We need to find the sum of all integers from 9 to 600.

1. **First term \( a \) = 9**
2. **Last term \( l \) = 600**
3. **Common difference \( d \) = 1**

To find the number of terms \( n \), use the formula for the nth term of an arithmetic series:

\[
l = a + (n - 1) \times d
\]

Substituting the values:

\[
600 = 9 + (n - 1) \times 1
\]

Solving for \( n \):

\[
600 - 9 = n - 1
\]
\[
n = 592
\]

Now, use the sum formula:

\[
S_{592} = \frac{592}{2} \times (9 + 600) = 296 \times 609 = 180,264
\]

So, the sum \( 9 + 10 + 11 + \cdots + 600 = 180,264 \).

### Part (b) - Sum from 9 to \( k \)

Here, the arithmetic series starts at 9 and ends at \( k \). We need to express the sum in a closed form.

1. **First term \( a \) = 9**
2. **Last term \( l = k \)**
3. **Common difference \( d = 1 \)**

The number of terms \( n \) is:

\[
n = k - 9 + 1 = k - 8
\]

Now, the sum can be expressed as:

\[
S_n = \frac{n}{2} \times (a + l) = \frac{k - 8}{2} \times (9 + k)
\]

So, the sum \( 9 + 10 + 11 + \cdots + k = \frac{k - 8}{2} \times (9 + k) \).

---

If you have any further questions or need additional details, feel free to ask!

### Relative Questions:
1. How would the solution change if the first term was 10 instead of 9?
2. What if the common difference \( d \) was 2 instead of 1?
3. Can you derive the sum formula for arithmetic series using induction?
4. How would you calculate the sum if the last term was a variable \( n \) rather than a constant?
5. How can you verify the result using another method or formula?

### Tip:
Always double-check your formulas and calculations, especially when dealing with large sums or algebraic expressions.

Learn how to find the sum of an arithmetic series from 9 to 600 and express the sum from 9 to k in closed form using arithmetic progression formulas. Suitable for high school students in Grades 9-12, this problem covers the sum formulas and step-by-step calculations.

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