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.