find the sum of all the even integers from 12 to 864

To find the sum of all even integers from 12 to 864, we can use the formula for the sum of an arithmetic sequence:

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

Where:
- \(S_n\) is the sum of the arithmetic sequence.
- \(n\) is the number of terms.
- \(a\) is the first term (12 in this case).
- \(l\) is the last term (864 in this case).

### Step 1: Determine the number of terms, \(n\)

The sequence of even numbers from 12 to 864 forms an arithmetic sequence where the common difference, \(d = 2\). To find \(n\), the number of terms, we use the formula for the \(n\)-th term of an arithmetic sequence:

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

Solving for \(n\):

\[
864 = 12 + (n - 1) \cdot 2
\]
\[
864 - 12 = (n - 1) \cdot 2
\]
\[
852 = (n - 1) \cdot 2
\]
\[
n - 1 = \frac{852}{2}
\]
\[
n - 1 = 426
\]
\[
n = 426 + 1 = 427
\]

### Step 2: Calculate the sum

Now that we know \(n = 427\), \(a = 12\), and \(l = 864\), we can calculate the sum:

\[
S_n = \frac{427}{2} \cdot (12 + 864)
\]
\[
S_n = \frac{427}{2} \cdot 876
\]
\[
S_n = 213.5 \cdot 876
\]
\[
S_n = 187,446
\]

Thus, the sum of all even integers from 12 to 864 is **187,446**.

Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. How would the sum change if we included odd integers as well?
2. What if the range was from 10 to 860? How would the sum differ?
3. How can we derive the formula for the sum of an arithmetic sequence?
4. What is the sum of all integers between 12 and 864?
5. What is the average of all even numbers between 12 and 864?

**Tip:** For arithmetic sequences, once you know the first term, last term, and number of terms, finding the sum is straightforward using the sum formula.

Find the sum of all even integers from 12 to 864 using the arithmetic sequence sum formula. This detailed solution covers how to calculate the number of terms, apply the sequence sum formula, and solve for the total sum of 187,446. Suitable for students in Grades 9-11.

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