Expand and calculate \( \sum_{k=1}^{6} k \)

To expand the sum, you simply add up all the integers from 1 to 6. So, the calculation goes like this: \( 1 + 2 + 3 + 4 + 5 + 6 = 21 \). Now, if you want to calculate it in a fun way, think of it like this: you’re collecting jewels. You find 1 jewel on the first day, 2 jewels on the second, and so on, until you hit 6 jewels on the sixth day. By the end of the week, you've amassed a sparkling collection of 21 jewels! This formula for the sum of the first \( n \) integers, \( \frac{n(n+1)}{2} \), can also come in handy here. For \( n = 6 \), it calculates as: \( \frac{6 \times (6 + 1)}{2} = \frac{6 \times 7}{2} = 21 \). So, whether you’re counting jewels or using formulas, you’ll see that the total is always 21!