Question 14 (1 point) Saved Evaluate 2^2+4^2+6^2+8^2ldots +50^2 infty 22000 22100 A 21100 21000

The given expression is the sum of the squares of even numbers from 2 to 50. We can write this as:<br /><br />$\sum_{n=1}^{25} (2n)^2 = \sum_{n=1}^{25} 4n^2 = 4\sum_{n=1}^{25} n^2$<br /><br />We know the formula for the sum of the first $k$ squares is:<br /><br />$\sum_{n=1}^{k} n^2 = \frac{k(k+1)(2k+1)}{6}$<br /><br />In our case, $k = 25$. So,<br /><br />$\sum_{n=1}^{25} n^2 = \frac{25(25+1)(2(25)+1)}{6} = \frac{25(26)(51)}{6} = \frac{25(26)(51)}{6} = 25(13)(17) = 5525$<br /><br />Now, multiply by 4:<br /><br />$4\sum_{n=1}^{25} n^2 = 4(5525) = 22100$<br /><br />So, the sum is 22100.<br /><br />Final Answer: The final answer is $\boxed{22100}$