What is the sum of all even numbers between 1 and 51?
indicate and demonstrate simple step by step formula. 

To find the sum of all even numbers between 1 and 51, we can use a simple mathematical formula for the sum of an arithmetic sequence.

### Step-by-Step Solution:

1. **Identify the even numbers between 1 and 51:**
   The even numbers between 1 and 51 are: 2, 4, 6, ..., 50.

2. **Identify the first term (a) and the last term (l) of the sequence:**
   - First even number \( a = 2 \)
   - Last even number \( l = 50 \)

3. **Find the number of terms (n):**
   The difference between consecutive even numbers is 2, and the sequence starts at 2 and ends at 50.
   
   To find the number of terms in the sequence, use the formula:
   \[
   n = \frac{l - a}{\text{common difference}} + 1
   \]
   Substituting values:
   \[
   n = \frac{50 - 2}{2} + 1 = \frac{48}{2} + 1 = 24 + 1 = 25
   \]
   So, there are 25 even numbers between 1 and 51.

4. **Use the formula for the sum of an arithmetic sequence:**
   The sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is given by:
   \[
   S_n = \frac{n}{2} \times (a + l)
   \]
   Substituting values:
   \[
   S_{25} = \frac{25}{2} \times (2 + 50) = \frac{25}{2} \times 52 = 25 \times 26 = 650
   \]

Thus, the sum of all even numbers between 1 and 51 is **650**.

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### Do you want any further details or have questions?

### Related questions:
1. How would the formula change if you sum odd numbers instead of even numbers?
2. How do you find the sum of even numbers between any two given numbers?
3. What is the sum of all odd numbers between 1 and 51?
4. Can you find the sum of all multiples of 3 between 1 and 100?
5. How does the arithmetic sequence formula work for a geometric progression?

### Tip: 
When summing numbers that follow a regular pattern (like even numbers), the arithmetic sequence formula can greatly simplify your calculations.

What is the sum of all even numbers between 1 and 51? Indicate and demonstrate a simple step-by-step formula.

Learn how to find the sum of all even numbers between 1 and 51 using the arithmetic sequence sum formula. This step-by-step guide includes key concepts such as identifying terms and applying the summation formula. Ideal for students in Grades 6-8.

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