https://file.aiquickdraw.com/mathbot/file/1727410229301eaxfnggo.jpg

The image shows a series of summation problems with instructions written in Indonesian, asking the user to "Calculate the sums of the following."

Here are the problems listed:

1. \( \sum_{k=2}^{7} 3k \)
2. \( \sum_{k=2}^{8} (2k + 3) \)
3. \( \sum_{i=2}^{5} (-1)^i \)
4. \( \sum_{n=3}^{7} (2 - n) \)

Let's go through and solve each summation step by step.

### 1. \( \sum_{k=2}^{7} 3k \)
The sum involves calculating \( 3k \) for \( k = 2, 3, 4, 5, 6, 7 \).

- \( 3(2) = 6 \)
- \( 3(3) = 9 \)
- \( 3(4) = 12 \)
- \( 3(5) = 15 \)
- \( 3(6) = 18 \)
- \( 3(7) = 21 \)

Adding these up:
\[ 6 + 9 + 12 + 15 + 18 + 21 = 81 \]

Thus, the sum is **81**.

### 2. \( \sum_{k=2}^{8} (2k + 3) \)
The sum involves calculating \( 2k + 3 \) for \( k = 2, 3, 4, 5, 6, 7, 8 \).

- \( 2(2) + 3 = 4 + 3 = 7 \)
- \( 2(3) + 3 = 6 + 3 = 9 \)
- \( 2(4) + 3 = 8 + 3 = 11 \)
- \( 2(5) + 3 = 10 + 3 = 13 \)
- \( 2(6) + 3 = 12 + 3 = 15 \)
- \( 2(7) + 3 = 14 + 3 = 17 \)
- \( 2(8) + 3 = 16 + 3 = 19 \)

Adding these up:
\[ 7 + 9 + 11 + 13 + 15 + 17 + 19 = 91 \]

Thus, the sum is **91**.

### 3. \( \sum_{i=2}^{5} (-1)^i \)
This summation alternates between \( (-1)^i \) for \( i = 2, 3, 4, 5 \).

- \( (-1)^2 = 1 \)
- \( (-1)^3 = -1 \)
- \( (-1)^4 = 1 \)
- \( (-1)^5 = -1 \)

Adding these up:
\[ 1 + (-1) + 1 + (-1) = 0 \]

Thus, the sum is **0**.

### 4. \( \sum_{n=3}^{7} (2 - n) \)
The sum involves calculating \( 2 - n \) for \( n = 3, 4, 5, 6, 7 \).

- \( 2 - 3 = -1 \)
- \( 2 - 4 = -2 \)
- \( 2 - 5 = -3 \)
- \( 2 - 6 = -4 \)
- \( 2 - 7 = -5 \)

Adding these up:
\[ -1 + (-2) + (-3) + (-4) + (-5) = -15 \]

Thus, the sum is **-15**.

---

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

### Related Questions:
1. How do we calculate the sum of an arithmetic series using a formula?
2. Can you explain how alternating sums work in general?
3. What is the difference between summation notation and a product notation?
4. How can we simplify summation problems involving constants?
5. How would these problems change if the bounds of summation were negative?

### Tip:
When evaluating summations, it is helpful to write out the first few terms to identify patterns or simplify calculations.

Hitunglah jumlah-jumlah berikut:
1. \( \sum_{k=2}^{7} 3k \)
2. \( \sum_{k=2}^{8} (2k + 3) \)
3. \( \sum_{i=2}^{5} (-1)^i \)
4. \( \sum_{n=3}^{7} (2 - n) \)

Learn how to solve summation problems like \( \sum_{k=2}^{7} 3k \) and \( \sum_{i=2}^{5} (-1)^i \), including detailed step-by-step solutions for arithmetic series and alternating sums. Suitable for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation