The formula to calculate the sum of the squares of the first n natural numbers is [n(n+1)(2n+1)] / 6. This formula provides a straightforward method to determine the sum without having to manually square and add each number.
According to the reference, the "Sum of squares of first n natural numbers means sum of the squares of the given series of natural numbers" and that "Sum of squares of n natural numbers can be calculated using the formula [n(n+1)(2n+1)] / 6". This confirms the correctness of the formula and provides context for its application.
Let's explore this formula further:
Understanding the Formula
- n: This represents the number of natural numbers you are considering. For instance, if you want the sum of the squares of the first 5 natural numbers, n would be 5.
- n+1: This term adds one to the value of n.
- 2n+1: This term multiplies n by two and adds one.
- [n(n+1)(2n+1)]: This part multiplies n, n+1, and 2n+1 together.
- / 6: The result of the previous multiplication is then divided by 6.
Example:
Let's find the sum of the squares of the first 3 natural numbers (1, 2, and 3).
- n = 3
- n + 1 = 3 + 1 = 4
- 2n + 1 = 2 * 3 + 1 = 7
- [n(n+1)(2n+1)] = 3 4 7 = 84
- 84 / 6 = 14
Therefore, 12 + 22 + 32 = 1 + 4 + 9 = 14, confirming the formula's accuracy.
Application:
This formula is not just a mathematical curiosity. It has practical applications in various fields such as:
- Physics: Calculating moments of inertia in rotational mechanics.
- Statistics: Calculating variance and standard deviation.
- Computer Science: Analyzing the time complexity of certain algorithms.
Summary
The formula [n(n+1)(2n+1)] / 6 provides an efficient way to find the sum of the squares of the first n natural numbers, saving both time and effort, as stated in the provided reference. This single formula is key to numerous mathematical and scientific computations, proving its practical significance.
