(2n%2B1))%2F6.jpeg "(n(n+1)(2n+1))/6")
The Formula for the Sum of Squares: (n(n+1)(2n+1))/6
The formula (n(n+1)(2n+1))/6 represents a powerful tool in mathematics. It allows us to directly calculate the sum of the squares of the first n natural numbers. Let's explore this formula in detail, understanding its derivation, applications, and significance.
Understanding the Formula
The formula states that the sum of squares of the first n natural numbers, which can be represented as:
1² + 2² + 3² + ... + n²
is equal to (n(n+1)(2n+1))/6.
Derivation
The derivation of this formula is usually done using mathematical induction.
-
Base Case: For n = 1, the formula holds true as 1² = (1(1+1)(2*1+1))/6 = 1.
-
Inductive Hypothesis: Assume the formula holds true for some arbitrary value of n = k. This means 1² + 2² + ... + k² = (k(k+1)(2k+1))/6.
-
Inductive Step: We need to prove that the formula also holds for n = k+1.
- Consider the sum of squares up to k+1: 1² + 2² + ... + k² + (k+1)²
- Using the inductive hypothesis, we can substitute (k(k+1)(2k+1))/6 for the sum of squares up to k.
- This gives us: (k(k+1)(2k+1))/6 + (k+1)²
- Simplifying the expression, we get: ((k+1)(k(2k+1) + 6(k+1)))/6
- Further simplification leads to: ((k+1)(2k²+7k+6))/6 = ((k+1)(k+2)(2k+3))/6
- This expression matches the original formula for n = k+1.
Therefore, the formula holds true for all natural numbers n based on the principle of mathematical induction.
Applications
The formula for the sum of squares has numerous applications across various fields, including:
- Statistics: Used in calculating the variance and standard deviation of data sets.
- Physics: Applies in problems involving motion and energy.
- Computer Science: Useful in analyzing algorithms and data structures.
Significance
The formula (n(n+1)(2n+1))/6 simplifies the calculation of the sum of squares, making it a valuable tool for mathematicians, scientists, and engineers. It highlights the beauty of mathematical patterns and the power of inductive reasoning.
Conclusion
The formula for the sum of squares is a fundamental result in mathematics with far-reaching applications. It serves as a testament to the elegance and power of mathematical formulas and the value of inductive reasoning in proving their validity.