https://file.aiquickdraw.com/mathbot/file/1728551224940zihlsbh6.jpg

Let's compute the sums based on the given images.

### Problem 1:

We are asked to compute the sum:

\[
\sum_{i=1}^{3} (2i - 1)
\]

This means we need to evaluate the expression \( 2i - 1 \) for each integer \( i \) from 1 to 3, and then sum the results:

1. For \( i = 1 \): \( 2(1) - 1 = 2 - 1 = 1 \)
2. For \( i = 2 \): \( 2(2) - 1 = 4 - 1 = 3 \)
3. For \( i = 3 \): \( 2(3) - 1 = 6 - 1 = 5 \)

Now, sum these results:

\[
1 + 3 + 5 = 9
\]

Thus, the value of the sum is **9**.

---

### Problem 2:

The sum to compute is:

\[
\sum_{k=-1}^{1} (k^3 + 2)
\]

This means we need to evaluate \( k^3 + 2 \) for \( k = -1, 0, 1 \) and sum the results:

1. For \( k = -1 \): \( (-1)^3 + 2 = -1 + 2 = 1 \)
2. For \( k = 0 \): \( 0^3 + 2 = 0 + 2 = 2 \)
3. For \( k = 1 \): \( 1^3 + 2 = 1 + 2 = 3 \)

Now, sum these results:

\[
1 + 2 + 3 = 6
\]

Thus, the value of the sum is **6**.

---

Would you like further details or have any questions?

Here are five related questions you could explore:
1. How would the sum change in Problem 1 if the upper limit was 4 instead of 3?
2. What is the general formula for the sum of an arithmetic series like in Problem 1?
3. In Problem 2, how does changing the lower or upper limit of the summation affect the result?
4. What happens to the sum in Problem 2 if you change the expression to \( k^2 + 2 \)?
5. How can sigma notation be used to represent other types of series, such as geometric series?

**Tip:** When working with sigma notation, always make sure to evaluate each term individually before summing, as small mistakes in one term can affect the entire result!

Compute the sums: 1. Σ(2i - 1) from i = 1 to 3, 2. Σ(k^3 + 2) from k = -1 to 1.

Learn how to compute summation problems including Σ(2i - 1) from i=1 to 3 and Σ(k^3 + 2) from k=-1 to 1 with detailed step-by-step solutions. This problem covers key algebraic concepts and summation techniques, suitable for students in Grades 8-10.

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