Find the sum of the first 10 even numbers.

To find the sum of the first 10 even numbers, we can start by identifying what those numbers are. Even numbers are integers that are divisible by 2, so the first 10 even numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. Now, let's add them together step by step.

Listing the Even Numbers

The first 10 even numbers can be listed as follows:

• 2
• 4
• 6
• 8
• 10
• 12
• 14
• 16
• 18
• 20

Calculating the Sum

Now, let's add these numbers together:

• 2 + 4 = 6
• 6 + 6 = 12
• 12 + 8 = 20
• 20 + 10 = 30
• 30 + 12 = 42
• 42 + 14 = 56
• 56 + 16 = 72
• 72 + 18 = 90
• 90 + 20 = 110

So, the sum of the first 10 even numbers is 110.

Using a Formula for Efficiency

Instead of adding each number individually, we can also use a formula to find the sum of the first n even numbers. The formula is:

Sum = n × (n + 1)

In this case, n is 10 (since we are looking for the first 10 even numbers). Plugging in the value:

Sum = 10 × (10 + 1) = 10 × 11 = 110

Understanding the Concept

This formula works because the first n even numbers can be expressed as 2, 4, 6, ..., 2n. When you factor out the 2, you get:

2 × (1 + 2 + 3 + ... + n)

The sum of the first n natural numbers (1 + 2 + 3 + ... + n) is given by the formula:

Sum = n × (n + 1) / 2

Thus, the sum of the first n even numbers becomes:

2 × (n × (n + 1) / 2) = n × (n + 1)

Final Thoughts

Whether you add the numbers directly or use the formula, the sum of the first 10 even numbers is consistently found to be 110. This approach not only saves time but also reinforces your understanding of arithmetic sequences and series. If you have any more questions or need further clarification, feel free to ask!