Question

FINDING A PATTERN In this exercise, you will explore the sequence of square numbers. The first four square numbers are represented below. 1 a. Find the differences between consecutive square numbers. Explain what you notice. b. Show how the polynomial identity \((n+1)^2-n^2=2 n+1\) models the differences between square numbers. c. Prove the polynomial identity in part (b).

Short answer

The differences between consecutive square numbers are successive odd numbers: 3, 5, 7, .... The polynomial identity \((n+1)^2-n^2=2n+1\) models these differences. The correctness of this identity is proven by expanding \((n+1)^2\) to obtain \(n^2 + 2n + 1\). When subtracting \(n^2\), the remaining is \(2n + 1\) which equals the right hand side of the identity.

Step-by-step solution

1: Find the Differences Between Consecutive Square Numbers

The first four square numbers are \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), and \(4^2 = 16\). The differences between these numbers are \(4-1=3\), \(9-4=5\), and \(16-9=7\). You can notice that the differences are successive odd numbers: 3, 5, and 7.

2: Show how the Polynomial Identity Models the Differences

The polynomial identity given is \((n+1)^2 - n^2 = 2n+1\). If we substitute values of \(n\), we can see that it corresponds to the difference between successive squares. For instance, when \(n=1: (1+1)^2 - 1^2 = 4-1 = 3\), it's the difference between the squares \(2^2\) and \(1^2\). This is the same result we got in the first step, hence the given polynomial identity models the differences between the square numbers.

3: Prove the Polynomial Identity

The given identity is \((n+1)^2 - n^2 = 2n+1\).Expanding \((n+1)^2\) gives \(n^2 + 2n + 1\).So, the left-hand side of the equation: \((n+1)^2 -n^2 = (n^2 + 2n + 1) -n^2 = 2n + 1\). As a result, we've shown that \((n+1)^2 - n^2 = 2n+1\).

Key concepts

Square Numbers

Square numbers hold a special place in mathematics, often visualized as the areas of perfect squares. A square number results from multiplying a number by itself. For example, the square of 4, written as 4^2, equals 16.

Understanding square numbers is essential for exploring patterns in arithmetic and algebra. The exercise provided asks us to investigate the differences between consecutive square numbers, such as 1, 4, 9, and so on. Upon inspection, one realizes that the progression from one square number to the next involves adding consecutive odd numbers, which is a fascinating and consistent pattern.

This pattern isn't just a curiosity; it has implications for understanding number theory and can assist in solving complex mathematical problems.

Difference of Squares

The difference of squares is an algebraic identity that demonstrates a unique relationship between square numbers. It states that for any number n, the difference between the squares of (n + 1) and n can be expressed as (n+1)^2 - n^2. This identity simplifies to 2n + 1, which reveals why the differences between consecutive squares are always odd numbers.

Observing the difference of squares gives us insight into how numbers build upon one another to form patterns. In the given exercise, we are shown that each leap from one square number to the next produces these successively larger odd numbers, a realization that can help to predict and calculate values within numerical sequences.

Consecutive Numbers

Consecutive numbers are numbers that follow each other in order, with a difference of one between each pair. In our context, we discuss consecutive square numbers which follow the pattern 1^2, 2^2, 3^2, 4^2, and so on. The concept of consecutive numbers is crucial for understanding sequences and series in mathematics.

When working with consecutive square numbers, the differences between them are of particular interest. As demonstrated in the exercise, these differences (3, 5, 7...) are not just any numbers, but specifically consecutive odd numbers. This discovery aids in recognizing patterns and formulating equations that describe the behavior of square numbers within the number system.

Mathematical Proofs

Mathematical proofs are logical arguments that establish the truth of a mathematical statement beyond doubt. These proofs rely on deductive reasoning and previously established theorems or axioms. In our exercise, proving the polynomial identity (n+1)^2 - n^2 = 2n + 1 involves expanding the squared terms and simplifying the equation.

The proof serves as a conclusive verification that the pattern observed empirically in the differences of consecutive square numbers is, in fact, a definitive mathematical relationship. This process of proof is fundamental in all areas of mathematics, as it cements our understanding and guarantees that our conclusions follow logically from sound principles.