To evaluate the sum $\sum_{k=1}^{8} (-5k)$, we need to compute the individual terms of the sum and then add them together. The summation means we are adding the expression $-5k$ for each value of $k$ from 1 to 8.

So let's break it down:

$\sum_{k=1}^{8} (-5k) = (-5 \cdot 1) + (-5 \cdot 2) + (-5 \cdot 3) + (-5 \cdot 4) + (-5 \cdot 5) + (-5 \cdot 6) + (-5 \cdot 7) + (-5 \cdot 8)$

Now, calculate each term:

$-5 \cdot 1 = -5$ $-5 \cdot 2 = -10$ $-5 \cdot 3 = -15$ $-5 \cdot 4 = -20$ $-5 \cdot 5 = -25$ $-5 \cdot 6 = -30$ $-5 \cdot 7 = -35$ $-5 \cdot 8 = -40$

Now, sum them up:

$-5 + (-10) + (-15) + (-20) + (-25) + (-30) + (-35) + (-40)$

$= -5 - 10 - 15 - 20 - 25 - 30 - 35 - 40$

$= -180$

Thus, the value of the sum is:

$\sum_{k=1}^{8} (-5k) = -180$

Would you like more details on the steps, or have any questions?

Here are some related questions:

1. What is the general formula for an arithmetic series?
2. How can we factor constants out of summation notation?
3. What would the sum be if the range changed to 1 to 10?
4. How does the sum of negative terms behave compared to positive ones?
5. Can we represent the sum $\sum_{k=1}^{8} (-5k)$ as a product of two terms?

Tip: Always check if a summation can be simplified using known formulas for arithmetic or geometric series.