The sum of the first $n$ square numbers is given by the formula:

$S_n = \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$

Explanation:

This formula is derived using mathematical induction or through summation techniques in algebra.

Would you like a proof or further explanation?

Related Questions:

1. How do you prove the formula using mathematical induction?
2. What is the sum of the first 100 square numbers?
3. How does this formula relate to the formula for the sum of the first $n$ natural numbers?
4. Can this formula be derived using calculus techniques like integration?
5. How can we use this formula in real-world applications, such as physics or finance?

Tip:

For large values of $n$, you can approximate the sum using integration:
$\int_1^n x^2 dx = \frac{n^3}{3}$ which gives an approximation to $S_n$.