2. \( \sum_{n=2}^{5} n^{2} \)

To calculate \( \sum_{n=2}^{5} n^{2} \), we simply need to square each integer from 2 to 5 and then sum them up. So, we have: \[ 2^{2} + 3^{2} + 4^{2} + 5^{2} = 4 + 9 + 16 + 25. \] Adding these together gives \( 4 + 9 = 13 \), \( 13 + 16 = 29 \), and \( 29 + 25 = 54 \). Thus, \( \sum_{n=2}^{5} n^{2} = 54 \). Need a fun fact? The formula for the sum of squares of the first \( n \) natural numbers is \( \frac{n(n+1)(2n+1)}{6} \). While it can seem daunting, this elegant formula could have saved you some squaring and summing! In real life, summing squares often pops up in statistics, particularly in calculating variance where the squares of deviations from the mean are summed up. It's a fundamental concept that helps us understand how spread out our data is!