https://r2.erweima.ai/mathbot/image/17262688556603b9cz8gh.jpg

The given expression is:

\[
\sum_{k=10}^{25} (2k) = 560
\]

This is a summation where \(k\) ranges from 10 to 25, and the general term in the summation is \(2k\).

### Step-by-step explanation:

1. **Understanding the Summation**:
   The expression means that we are summing up the values of \(2k\) for all integers \(k\) from 10 to 25. This can be expanded as:
   
   \[
   2(10) + 2(11) + 2(12) + \dots + 2(25)
   \]

2. **Factoring out the constant 2**:
   Since each term has a factor of 2, we can factor it out of the summation:

   \[
   2 \sum_{k=10}^{25} k
   \]

3. **Sum of integers from 10 to 25**:
   The summation of integers from 10 to 25 can be calculated using the formula for the sum of an arithmetic series:

   \[
   S = \frac{n}{2} \times (\text{first term} + \text{last term})
   \]

   Here, the first term is 10, the last term is 25, and \(n\) is the number of terms. To find \(n\), subtract 10 from 25 and add 1:

   \[
   n = 25 - 10 + 1 = 16
   \]

   Now calculate the sum:

   \[
   S = \frac{16}{2} \times (10 + 25) = 8 \times 35 = 280
   \]

4. **Multiplying by 2**:
   Now, multiply the sum by 2 to get the final result:

   \[
   2 \times 280 = 560
   \]

Thus, the summation evaluates to 560, which matches the given result in the image.

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Would you like more details or have any further questions? Here are five related questions that might help:

1. How would the sum change if \(k\) ranged from 15 to 30?
2. Can you explain how to calculate the sum of an arithmetic series?
3. What is the general formula for summing an arithmetic progression?
4. How can we derive the formula for the sum of a sequence like this one?
5. How would the result change if the terms were of the form \(3k\) instead of \(2k\)?

**Tip**: For summing arithmetic sequences, always check if you can factor out constants to simplify the calculation!

Calculate the summation: ∑(2k) from k = 10 to k = 25, and the result is 560.

This problem explains how to calculate the summation ∑(2k) from k = 10 to k = 25, using the arithmetic series formula. Learn how to factor out constants and find the sum of integers in this detailed guide, suitable for Grades 9-12.

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