Find the sum of the first 50 even natural numbers.

To find the sum of the first 50 even natural numbers, we can use a straightforward approach based on the properties of even numbers. Even natural numbers are simply the multiples of 2, so the first 50 even natural numbers can be expressed as: 2, 4, 6, ..., 100. This sequence can also be represented as 2 times the first 50 natural numbers: 2 × 1, 2 × 2, 2 × 3, ..., 2 × 50.

Understanding the Sequence

The first 50 even natural numbers can be written as:

• 2 × 1 = 2
• 2 × 2 = 4
• 2 × 3 = 6
• ...
• 2 × 50 = 100

Using the Formula for the Sum of Natural Numbers

To find the sum of these even numbers, we can use the formula for the sum of the first n natural numbers, which is:

S = n(n + 1) / 2

In this case, n is 50. So, we first calculate the sum of the first 50 natural numbers:

S = 50(50 + 1) / 2 = 50 × 51 / 2 = 1275

Calculating the Sum of Even Numbers

Since each even number is double its corresponding natural number, the sum of the first 50 even natural numbers will be:

Sum of even numbers = 2 × S

Substituting the value we found:

Sum of even numbers = 2 × 1275 = 2550

Final Result

Therefore, the sum of the first 50 even natural numbers is 2550.

Visualizing the Calculation

To visualize this, think of pairing the numbers:

• 2 + 100 = 102
• 4 + 98 = 102
• 6 + 96 = 102
• ...
• 50 pairs in total

Each pair sums to 102, and since there are 25 pairs, we can also calculate:

25 × 102 = 2550

This method not only confirms our earlier calculation but also provides a different perspective on how the sum is derived. Thus, the answer remains consistent: the sum of the first 50 even natural numbers is indeed 2550.