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.