Let's expand and solve the sigma notation problems shown in the image step by step:

1. $\sum_{i=2}^{8} \sqrt{i} + x$

   This is a sum of square roots of integers from 2 to 8, plus the constant $x$. The constant $x$ is added at each step but doesn't depend on $i$.

   $\sum_{i=2}^{8} \sqrt{i} + x = (\sqrt{2} + x) + (\sqrt{3} + x) + (\sqrt{4} + x) + (\sqrt{5} + x) + (\sqrt{6} + x) + (\sqrt{7} + x) + (\sqrt{8} + x)$

   Simplifying: $= (\sqrt{2} + \sqrt{3} + \sqrt{4} + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8}) + 7x$

2. $\sum_{i=1}^{5} \frac{1}{i}$

   This is a harmonic sum from 1 to 5: $\sum_{i=1}^{5} \frac{1}{i} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}$ Calculating the terms: $= 1 + 0.5 + 0.3333 + 0.25 + 0.2 = 2.2833$

3. $\sum_{i=7}^{10} (x - 2)^i$

   This is a sum of powers of $(x-2)$ from $i = 7$ to $i = 10$: $\sum_{i=7}^{10} (x - 2)^i = (x - 2)^7 + (x - 2)^8 + (x - 2)^9 + (x - 2)^{10}$ The result will remain in this form unless the value of $x$ is known, making it impossible to simplify further without additional information.

4. $\sum_{i=2}^{6} (x_i + y_i)$

   This is a sum of terms involving $x_i$ and $y_i$ from $i = 2$ to $i = 6$: $\sum_{i=2}^{6} (x_i + y_i) = (x_2 + y_2) + (x_3 + y_3) + (x_4 + y_4) + (x_5 + y_5) + (x_6 + y_6)$ Without values for $x_i$ and $y_i$, this is the simplest expression possible.

5. $\sum_{i=1}^{6} m^i + a$

   This is a sum of powers of $m$ from 1 to 6, with the constant $a$ added to the entire sum: $\sum_{i=1}^{6} m^i + a = (m^1 + m^2 + m^3 + m^4 + m^5 + m^6) + a$ Simplifying: $= (m + m^2 + m^3 + m^4 + m^5 + m^6) + a$

Would you like more detailed explanations for any specific steps?

Here are 5 related questions:

1. How would you calculate the value of a sigma notation when the terms include variables like $x$ and $m$?
2. What is the significance of the harmonic series, and how does it relate to $\sum_{i=1}^{n} \frac{1}{i}$?
3. How can the sum of squares or powers of expressions be simplified or solved?
4. What are the applications of sigma notation in real-world problems?
5. How does sigma notation simplify the representation of large sums?

Tip: When dealing with sigma notation, always check whether the terms involve constants or variables, as this affects how you simplify and compute the sum.