Sum of the squares of the first n natural numbers ∑r2
Sum of the squares of the first n natural numbers ∑r2
Sum of the Squares of the First n Natural Numbers ∑r²
Definition
- The sum of the squares of the first n natural numbers ∑r² is the sum obtained by squaring each natural number up to n and adding them all together.
- In mathematical notation, we say that ∑r² = 1² + 2² + 3² + … + n².
Formula
- The formula for the sum of the squares of the first n natural numbers ∑r² is given by n(n + 1)(2n + 1) / 6.
- This can also be written as (n/6) * (n + 1) * (2n + 1) for clarity in computation.
Derivation
- The formula can be derived using induction, where you assume the formula works for a given number k, then show it also works for the next number k + 1.
- It can also be derived by expressing each square number as a series of odd numbers, then recognising the summation as a sum of arithmetic series.
Application
- This formula is useful in various areas of mathematics and physics, especially when dealing with quadratic sums.
- An understanding of this formula is important for further algebraic manipulations which can occur in the Core Pure section of further maths.
- It can be used as a method to calculate the sum of squares in many practical scenarios which may occur in exam questions.
Examples
- For example, the sum of the squares of the first 3 natural numbers is 1² + 2² + 3² = 1+4+9 = 14.
- If we use the formula, n=3, so (3/6) * (3 + 1) * (2*3 + 1) = 14, which correlates with the manual summing.
Key Points
- The sum of the squares of the first n natural numbers ∑r² is determined by the formula n(n + 1)(2n + 1) / 6.
- This formula can be derived using a method called induction or by expressing each square number as a series of odd numbers.
- Knowledge of this formula is key in tackling algebraic computations in the Core Pure section of further maths and in real-world scenarios.