https://file.aiquickdraw.com/mathbot/file/1731725431683w5o9ml8a.jpg

This image presents a mathematical summation formula:

\[
\sum_{i=1}^n \left( 5i^2 - 2i \right) = \frac{1}{6} n(n+1)(10n - 1).
\]

### Breakdown:
1. **Left-hand Side**: This is the summation of the expression \(5i^2 - 2i\) as \(i\) varies from \(1\) to \(n\).
2. **Right-hand Side**: The closed-form formula \(\frac{1}{6} n(n+1)(10n - 1)\) simplifies the summation to a function of \(n\).

### Key Insights:
- Such summations often leverage standard formulas for the sums of powers of integers, such as:
  - \(\sum_{i=1}^n i = \frac{n(n+1)}{2}\)
  - \(\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}\)
- Using these standard formulas, the summation of \(5i^2 - 2i\) can be derived step by step.

Would you like me to detail the derivation of the right-hand side formula?

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#### Expanding this topic:
1. What are the formulas for summations of higher powers, like \(i^3\) or \(i^4\)?
2. How does this summation formula apply to real-world problems (e.g., in physics or computer science)?
3. How can this formula be verified for small values of \(n\)?
4. What are other techniques to simplify summations besides standard formulas?
5. How does this summation compare to other similar series, e.g., geometric progressions?

#### Tip:
Understanding summation formulas is crucial for solving complex series quickly. Always practice breaking down the terms systematically!

Evaluate the summation \(\sum_{i=1}^n \left( 5i^2 - 2i \right)\) and derive its closed-form formula.

Learn how to derive the closed-form formula for the summation \(\sum_{i=1}^n (5i^2 - 2i)\) using standard summation formulas for powers of integers. This example highlights key techniques in evaluating summations and simplifying polynomial series.

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