The image contains four summation problems in a sequence. Here's a transcription of each summation expression for clarity:

7. $\sum_{k=1}^{4} 3k$
8. $\sum_{k=2}^{9} (2k + 3)$
9. $\sum_{i=2}^{6} (-1)^i$
10. $\sum_{n=3}^{7} (2 - n)$

Let's solve each of these summations one by one:

1. Problem 7:

$\sum_{k=1}^{4} 3k = 3(1) + 3(2) + 3(3) + 3(4) = 3 + 6 + 9 + 12 = 30$

2. Problem 8:

$\sum_{k=2}^{9} (2k + 3) = (2(2) + 3) + (2(3) + 3) + \dots + (2(9) + 3)$ This evaluates to: $(4 + 3) + (6 + 3) + (8 + 3) + (10 + 3) + (12 + 3) + (14 + 3) + (16 + 3) + (18 + 3) = 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 = 112$

3. Problem 9:

$\sum_{i=2}^{6} (-1)^i = (-1)^2 + (-1)^3 + (-1)^4 + (-1)^5 + (-1)^6 = 1 - 1 + 1 - 1 + 1 = 1$

4. Problem 10:

$\sum_{n=3}^{7} (2 - n) = (2 - 3) + (2 - 4) + (2 - 5) + (2 - 6) + (2 - 7) = -1 - 2 - 3 - 4 - 5 = -15$

Final Answers:

• Problem 7: $30$
• Problem 8: $112$
• Problem 9: $1$
• Problem 10: $-15$

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. What is the general formula for summing an arithmetic sequence?
2. How do alternating sums like $(-1)^i$ behave as $i$ increases?
3. How can you apply summation properties to simplify large sums?
4. What patterns emerge in sums of linear functions like $2k + 3$?
5. How does shifting summation indices affect the result?

Tip: When dealing with summations, breaking down each term before adding can help simplify the process, especially when facing complex expressions.