Calculate the sum of all even natural numbers < 75

To calculate the sum of all even natural numbers less than 75, we can follow these steps:

1. **Identify the even numbers:** The even natural numbers less than 75 are 2, 4, 6, ..., 74.
   
2. **Determine the sequence:**
   - This is an arithmetic sequence where the first term \( a_1 \) is 2, the common difference \( d \) is 2, and the last term \( a_n \) is 74.

3. **Find the number of terms (n):**
   \[
   n = \frac{a_n - a_1}{d} + 1 = \frac{74 - 2}{2} + 1 = \frac{72}{2} + 1 = 36 + 1 = 37
   \]

4. **Calculate the sum of the sequence (S):**
   The sum \( S_n \) of an arithmetic sequence is given by:
   \[
   S_n = \frac{n}{2} \times (a_1 + a_n)
   \]
   Substituting the values:
   \[
   S_{37} = \frac{37}{2} \times (2 + 74) = \frac{37}{2} \times 76 = 37 \times 38 = 1406
   \]

So, the sum of all even natural numbers less than 75 is **1406**.

Would you like more details or have any questions?

Here are five related questions:
1. How would you calculate the sum of all odd natural numbers less than 75?
2. What is the sum of all even numbers between 1 and 100?
3. How can you find the sum of the first 50 even numbers?
4. How does the sum of even numbers differ from the sum of odd numbers in a similar range?
5. How would you find the sum of multiples of a different number, such as 3, under a certain limit?

**Tip:** The formula for the sum of an arithmetic sequence is a powerful tool for quickly calculating the sum of evenly spaced numbers.

Learn how to calculate the sum of all even natural numbers less than 75 with a step-by-step solution. This problem involves identifying even numbers, using the formula for the sum of an arithmetic sequence, and understanding basic arithmetic concepts. Suitable for students in Grades 7-9.

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