What is the sum of the first 100 even natural numbers?

The sum of the first 100 even natural numbers is 10100.

Detailed Explanation

The question asks for the sum of the first 100 even natural numbers. The even natural numbers are 2, 4, 6, 8, and so on. We need to find the sum:

2 + 4 + 6 + ... + 200

This is an arithmetic series. There are a couple of ways to solve this.

Method 1: Using the arithmetic series formula

The formula for the sum of an arithmetic series is:

S_n = n/2 * (a₁ + a_n)

Where:

• S_n is the sum of the first n terms
• n is the number of terms (in this case, 100)
• a₁ is the first term (in this case, 2)
• a_n is the last term (in this case, 200)

Plugging in the values:

S₁₀₀ = 100/2 (2 + 200)
S₁₀₀ = 50 202
S₁₀₀ = 10100

Method 2: Factoring out a 2

We can rewrite the sum as:

2(1 + 2 + 3 + ... + 100)

Now we need to find the sum of the first 100 natural numbers, which is given by the formula:

n(n+1)/2

In this case, n = 100, so the sum of the first 100 natural numbers is:

100(100+1)/2 = 100 101 / 2 = 50 101 = 5050

Multiplying this by 2 (from our earlier factoring):

2 * 5050 = 10100

Conclusion

Both methods yield the same result. According to the reference, the sum of the first 100 even numbers is indeed 10100.