sum i = 1 to 20 (x ^ 2 * sqrt((9x ^ 4)/(x ^ 8)) * sqrt(i ^ 4) + sqrt(125/5) * i ^ log_5(5) + log_3(243))

The given sum is expressed as:

\[
\sum_{i=1}^{20} \left( x^2 \cdot \sqrt{\frac{9x^4}{x^8}} \cdot \sqrt{i^4} + \sqrt{\frac{125}{5}} \cdot i^{\log_5(5)} + \log_3(243) \right)
\]

We need to break this down step by step.

1. **Simplifying \( \sqrt{\frac{9x^4}{x^8}} \):**
   \[
   \frac{9x^4}{x^8} = \frac{9}{x^4}
   \]
   So,
   \[
   \sqrt{\frac{9}{x^4}} = \frac{3}{x^2}
   \]

2. **Simplifying \( \sqrt{i^4} \):**
   \[
   \sqrt{i^4} = i^2
   \]

3. **Simplifying \( \sqrt{\frac{125}{5}} \):**
   \[
   \frac{125}{5} = 25
   \]
   So,
   \[
   \sqrt{25} = 5
   \]

4. **Simplifying \( i^{\log_5(5)} \):**
   Since \( \log_5(5) = 1 \),
   \[
   i^{\log_5(5)} = i^1 = i
   \]

5. **Simplifying \( \log_3(243) \):**
   Since \( 243 = 3^5 \),
   \[
   \log_3(243) = 5
   \]

Now, substituting all these simplified expressions back into the original sum, we get:

\[
\sum_{i=1}^{20} \left( x^2 \cdot \frac{3}{x^2} \cdot i^2 + 5 \cdot i + 5 \right)
\]

This simplifies further to:

\[
\sum_{i=1}^{20} \left( 3i^2 + 5i + 5 \right)
\]

Now, let's compute the sum:

\[
\sum_{i=1}^{20} \left( 3i^2 + 5i + 5 \right)
\]

This sum can be split into three separate sums:

\[
3 \sum_{i=1}^{20} i^2 + 5 \sum_{i=1}^{20} i + \sum_{i=1}^{20} 5
\]

- The sum of squares \( \sum_{i=1}^{n} i^2 \) is given by:
  \[
  \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}
  \]

- The sum of integers \( \sum_{i=1}^{n} i \) is given by:
  \[
  \sum_{i=1}^{n} i = \frac{n(n+1)}{2}
  \]

- The sum of a constant \( 5 \times 20 \) is simply:
  \[
  5 \times 20 = 100
  \]

Now, substituting \( n = 20 \):

1. \( \sum_{i=1}^{20} i^2 = \frac{20 \cdot 21 \cdot 41}{6} \)
2. \( \sum_{i=1}^{20} i = \frac{20 \cdot 21}{2} \)
3. \( \sum_{i=1}^{20} 5 = 100 \)

Let's calculate these sums.The value of the sum \( \sum_{i=1}^{20} \left( 3i^2 + 5i + 5 \right) \) is \( 9760 \).

If you have any questions or need further clarification, feel free to ask!

Here are five related questions to explore further:
1. How would the result change if the upper limit of the sum was extended to 30?
2. What if the \( x \)-dependent term was different, such as \( x^3 \) instead of \( x^2 \)?
3. How can you compute similar sums efficiently using a programming language?
4. How do you derive formulas for summing higher powers of integers, such as \( i^3 \) or \( i^4 \)?
5. What applications do these summation techniques have in physics or engineering?

**Tip:** When working with summations, try breaking complex expressions into simpler parts, just as we did with separate sums here.

Learn how to compute the summation ∑(3i² + 5i + 5) from i=1 to 20 step-by-step with detailed explanations and formulas. This problem involves summation techniques, square roots, logarithms, and is suitable for advanced high school students or undergraduates.

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