https://file.aiquickdraw.com/mathbot/file/1726477733748dp7u1lms.jpg

The image shows a summation formula:

\[
\sum_{i=1}^{n} i^2
\]

This represents the sum of the squares of the first \(n\) integers. The closed-form formula for this summation is:

\[
\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}
\]

Would you like further details on how this formula is derived or explained? Let me know if you need any other clarification.

Here are five related questions to expand on this topic:
1. How does the formula for the sum of cubes differ from the sum of squares?
2. Can this formula be used for approximating larger values of \(n\)?
3. What is the significance of summation formulas in calculus and series?
4. How do we find similar formulas for higher powers of \(i\)?
5. How does the sum of squares relate to statistical concepts like variance?

**Tip**: In summation problems, understanding both the closed-form solution and the properties of the series can help you solve related problems efficiently.

Explore the summation of squares formula: \( \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \). This formula is key in understanding quadratic series and summation techniques. Suitable for students in Grades 10-12.

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