Let's evaluate each of these sums one by one. These are examples of constant summations, where each term in the sum is a constant value, so we can use a simple formula:

For a constant sum $\sum_{k=a}^b c$, the sum is equal to $c \times (b - a + 1)$, where $c$ is the constant term.

Solution

(a) $\sum_{k=1}^7 5$

• Here, the constant term $c = 5$, and $b - a + 1 = 7 - 1 + 1 = 7$.
• So, the sum is: $5 \times 7 = 35$

(b) $\sum_{k=1}^{300} 7$

• Here, the constant term $c = 7$, and $b - a + 1 = 300 - 1 + 1 = 300$.
• So, the sum is: $7 \times 300 = 2100$

(c) $\sum_{k=4}^{240} 30$

• Here, the constant term $c = 30$, and $b - a + 1 = 240 - 4 + 1 = 237$.
• So, the sum is: $30 \times 237 = 7110$

Final Answers

• (a) $35$
• (b) $2100$
• (c) $7110$

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

Here are some related questions for practice:

1. How would you evaluate $\sum_{k=1}^{10} 3$?
2. What is the sum $\sum_{k=5}^{15} 8$?
3. How does the sum $\sum_{k=10}^{20} 12$ compare if we change the constant term?
4. How would you handle a summation where the limits $a$ and $b$ are negative?
5. What if the summation included a variable term instead of a constant?

Tip: When dealing with constant sums, multiply the constant by the number of terms directly for a quick solution.