Problem 2 - Discrete Mathematics with Applications

Question

Prove that the sum of two consecutive triangular numbers is a perfect square.

Step-by-Step Solution

Answer

The sum of two consecutive triangular numbers T(n) and T(n+1) is given by the expression T(n) + T(n+1) = (n * (n + 1) / 2) + ((n + 1) * (n + 2) / 2). After simplifying, we find that the sum equals (n+1)(n+1), or (n+1)^2. Therefore, the sum of two consecutive triangular numbers is a perfect square, as required.

Step 1: Write the formulas for T(n) and T(n+1)

Use the formula for triangular numbers, n * (n + 1) / 2, to write the formulas for T(n) and T(n+1). They will be: T(n) = n * (n + 1) / 2 and T(n+1) = (n + 1) * (n + 2) / 2

Step 2: Add T(n) and T(n+1)

Sum the two consecutive triangular numbers: T(n) + T(n+1) = (n * (n + 1) / 2) + ((n + 1) * (n + 2) / 2) Now we need to simplify this expression and determine if it is a perfect square.

Step 3: Simplify the expression

Factor out the common term (n+1) from both fractions: T(n) + T(n+1) = (n+1) * (n / 2 + (n + 2) / 2) Now find a common denominator for the two fractions, which is 2: T(n) + T(n+1) = (n+1) * ((n + n + 2) / 2)

Step 4: Further simplify the expression

Perform the operations in the numerator: T(n) + T(n+1) = (n+1) * (2n + 2) / 2 Now, we can simplify by canceling out the factor of 2: T(n) + T(n+1) = (n+1) * (n+1)

Step 5: Prove the sum is a perfect square

Since the simplified expression is (n+1) * (n+1), which can be written as (n+1)^2, the sum of two consecutive triangular numbers is a perfect square, as required.

Key Concepts

Perfect Squares

A perfect square is the product of a whole number multiplied by itself. For instance, 16 is a perfect square because it can be expressed as 4 times 4, or \(4^2\). This notion is not only fundamental in algebra but also has practical applications in areas such as geometry, where it represents the area of a square.

In the context of triangular numbers, we are looking for a scenario where the sum of two consecutive triangular numbers forms a perfect square. As we know, each triangle number can be visualized as a triangle with dots, with each subsequent number adding another row of dots, forming a larger triangle. The exercise shows through algebraic manipulation that adding two such numbers—essentially creating a square of dots—will give us a perfect square number.

Mathematical Proofs

Mathematical proofs are logical arguments that establish the truth of a mathematical statement. They are built from axioms—statements accepted as true without proof—and previously proven theorems. Proofs can take various forms including direct, contrapositive, contradiction, and induction.

In our triangular number exercise, we applied a direct proof. Here we started with the known formulas for the n-th and (n+1)-th triangular numbers, added them, and simplified the expression. The result was an algebraic expression that clearly matched the form of a perfect square, \(n+1)^2\), hence directly proving our initial statement that the sum is indeed a perfect square.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. For example, \(n * (n + 1) / 2\) is the algebraic expression for the n-th triangular number. Algebraic expressions can often be simplified or manipulated to reveal more information about the relationships between variables.

In our solution, simplification helped transform the sum of two algebraic expressions representing triangular numbers into another expression that was clearly recognizable as a perfect square. This is crucial because it demonstrates how algebraic manipulation can be used to prove properties of numerical patterns, such as those found in sequences of triangular numbers.