Evaluate the sum

Summation from k equals 1 to 8 left parenthesis negative 5 k right parenthesis∑k=18(−5k).

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Part 1

Summation from k equals 1 to 8 left parenthesis negative 5 k right parenthesis∑k=18(−5k)equals=enter your response here

​(Simplify your​ answer.)

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.

Evaluate the sum: Summation from k equals 1 to 8 of (-5k). Simplify your answer.

Learn how to evaluate the sum of an arithmetic series from k=1 to 8 of (-5k). This problem covers key concepts such as summation notation and arithmetic sequences, with a step-by-step breakdown. Suitable for students in Grades 6-8.

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