https://file.aiquickdraw.com/mathbot/file/17265567257522vga1v30.jpg

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.

Expand the following sigma notations:
1. Σ (i=2 to 8) √i + x
2. Σ (i=1 to 5) 1 / (2a)
3. Σ (i=7 to 10) (x - 2)^i
4. Σ (i=2 to 6) (x_i + y_i)
5. Σ (i=1 to 6) m^i + a

This page provides detailed solutions to problems involving sigma notation, summing square roots, and powers. Learn how to expand summations such as Σ (i=2 to 8) √i + x and Σ (i=1 to 6) m^i + a with step-by-step explanations. Suitable for high school students in Grades 10-12.

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