Question

Show that for \(n=0,1,2\) the following is true: $$1^{2}+2^{2}+3^{2}+\cdots+n^{2}=n(n+1)(2 n+1) / 6$$

Short answer

The equation holds for \(n=0\), \(1\), and \(2\).

Step-by-step solution

Substitute n=0 into the Equation

For \(n=0\), the sum \(1^2 + 2^2 + 3^2 + \cdots + n^2\) simplifies to 0, since there are no terms to add. Substituting into the right-hand side of the given expression, we have \(0(0+1)(2(0)+1)/6 = 0\). Therefore, the equation holds when \(n=0\).

Check n=1

For \(n=1\), the sum is simply \(1^2 = 1\). The right-hand side becomes \(1(1+1)(2(1)+1)/6 = 1\cdot 2\cdot 3/6 = 1\), which matches the left side. Thus, the equation holds for \(n=1\).

Verify n=2

For \(n=2\), we need to compute the sum: \(1^2 + 2^2 = 1 + 4 = 5\). The formula gives \(2(2+1)(2(2)+1)/6 = 2\cdot 3\cdot 5/6 = 5\). This matches the computed sum, so the equation holds for \(n=2\).

Key concepts

Sum of Squares

The sum of squares is a mathematical concept where you add the squares of consecutive integers. It's not only a topic of great curiosity but also a key component of various mathematical fields, like statistics and algebra. In our exercise, we look at the sum of the squares of numbers from 1 up to a number \(n\). This is expressed as:

• \(1^2 + 2^2 + 3^2 + \cdots + n^2\)

The sum of squares has a specific formula:

• \(1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}\)

This formula lets us calculate the sum directly without having to manually add each squared term. To confirm this formula's accuracy, mathematicians often use mathematical proofs like induction, as we've seen in the solution. Such a formula becomes efficient as \(n\) gets large, allowing for quick computations of problems involving sums of squares.

Algebraic Proof

An algebraic proof involves demonstrating that a certain mathematical statement is always true using algebraic methods. It's like a logical argument where you start from known truths and apply algebraic manipulations. In this exercise, we've shown that the formula for the sum of squares is true using specific cases: \(n=0\), \(n=1\), and \(n=2\). Here are a few steps taken in an algebraic proof:

• Substitute specific values to test initial cases, such as \(n=0\), \(n=1\), and \(n=2\).
• Calculate both the left-hand side (direct sum of squares) and right-hand side (algebraic expression) for these values.
• If both sides are equal, the equation holds for those specific cases.

Algebraic proofs often start with specific examples before moving to more general cases. In complex proofs, mathematicians use further techniques such as induction to ensure the formula holds for all\(n\). This method essentially builds a strong foundation of certainty around the mathematical statement.

Discrete Mathematics

Discrete mathematics is a branch of mathematics dealing with structures that are fundamentally discrete rather than continuous. This means it focuses on distinct and separate elements, like the integers used in sums or sequences. The sum of squares is a classic problem in discrete mathematics, illustrating the values and interactions of integers. Key aspects include:

• Counting methods, such as finding the sum of a finite series.
• Mathematical reasoning, like logical deductions in proofs.
• Sequences and series, fundamental for understanding relationships between numbers.

The problem of summing squares is discrete because it involves distinct numbers and their interactions without involving fractions or decimals. Discrete mathematics is essential for computer science, cryptography, and algorithm design, where clear, non-continuous solutions are vital. Understanding these concepts prepares students for more advanced topics and helps sharpen their problem-solving skills, leveraging precise mathematical logic.