The sum of the squares of the first 20 even natural numbers is 11,480.
To arrive at this answer, we can use a formula. First, let's define what the first 20 even natural numbers are: 2, 4, 6, 8, ..., 40. We are asked to find the sum:
2² + 4² + 6² + ... + 40²
We can rewrite this as:
(21)² + (22)² + (23)² + ... + (220)²
This simplifies to:
2²(1² + 2² + 3² + ... + 20²)
Now, we can use the formula for the sum of the squares of the first n natural numbers, which is:
Sum = n(n+1)(2n+1) / 6
In our case, n = 20. So, the sum of the squares of the first 20 natural numbers is:
Sum = 20(20+1)(220+1) / 6 = 20 21 * 41 / 6 = 2870
Finally, we multiply this by 2² (which is 4):
4 * 2870 = 11480
Therefore, the sum of the squares of the first 20 even natural numbers is 11,480.
