Problem 40

Question

Let $t_{n}$ denote the $n$ th triangular number. Prove that $8 t_{n}+1$ is a perfect square.

Step-by-Step Solution

Verified

Answer

We can prove that $8t_n + 1$ is a perfect square by expressing the nth triangular number, $t_n$, as a formula: $t_n = \frac{n(n + 1)}{2}$. Substituting this formula into $8t_n + 1$ and expanding, we get $8t_n + 1 = 4n(n + 1) + 1 = 4n^2 + 4n + 1$. We can then rewrite this as a perfect square: $8t_n + 1 = (2n + 1)^2$. Thus, we have proven that $8t_n + 1$ is a perfect square for any $n$th triangular number $t_n$.

1Step 1: Express the nth triangular number as a formula

The nth triangular number, $t_n$, is the sum of the first $n$ positive integers. It can be expressed as a formula: \[t_n = 1 + 2 + 3 + \cdots + n = \frac{n(n + 1)}{2}\]

2Step 2: Substitute the formula for $t_n$ into the equation and expand

Now, let's substitute this formula for $t_n$ into the equation $8t_n + 1$: \[ 8t_n + 1 = 8\left(\frac{n(n + 1)}{2}\right) + 1 \] Expanding the equation, we get: \[ 8t_n + 1 = 4n(n + 1) + 1 \]

3Step 3: Transform the equation into a perfect square

To show that the equation $(8t_n + 1)$ is a perfect square, we can try to manipulate the expanded equation in Step 2 into the form of a square of a binomial. Let's rearrange the equation and factor: \[ 8t_n + 1 = 4n^2 + 4n + 1 \] Now, we can write this as: \[ 8t_n + 1 = (2n + 1)(2n + 1) \]

4Step 4: Verify the perfect square form

Observing the equation we derived in Step 3, we can see that: \[ 8t_n + 1 = (2n + 1)^2 \] This equation demonstrates that $8t_n + 1$ is a perfect square: it is equal to the square of the integer $(2n + 1)$. Therefore, we have proven that $8t_n + 1$ is a perfect square for any $n$th triangular number $t_n$.

Key Concepts

Perfect SquareMathematical ProofBinomial Expansion

Perfect Square

A perfect square is a number that results from squaring an integer. For example, numbers like 1, 4, 9, 16, and 25 are perfect squares because they are obtained by multiplying an integer by itself (1x1, 2x2, 3x3, etc.). When proving mathematical results, identifying a perfect square is important because it assures us that the number is formed by a whole number's square.
In the given problem, we start with the triangular numbers defined by $t_n = \frac{n(n + 1)}{2}$. The task is to prove that $8t_n + 1$ is a perfect square. By solving the expression $8t_n + 1$ analytically, it was shown that it equals $(2n + 1)^2$. Thus, it is a perfect square since $2n + 1$ is an integer for any positive integer $n$. This illustrates a foundational aspect of number theory, simplifying and identifying terms to reveal deeper properties.

Mathematical Proof

Mathematical proof involves demonstrating the truth of a statement using logical reasoning derived from accepted mathematical principles. Proofs can take many forms—algebraic manipulations, geometric interpretations, or other logical steps.
For our exercise, the objective was to prove that $8t_n + 1$ is a perfect square. The proof involved a clear and concise series of steps:

Express the triangular number $t_n$ in terms of $n$.
Substitute this expression in $8t_n + 1$.
Simplify this to reveal it as a square of a binomial.

These steps used algebraic manipulation, which is a common strategy in mathematical proofs, especially in number theory. By rewriting expressions and recognizing patterns, we establish the equality $8t_n + 1 = (2n + 1)^2$, proving it a perfect square. This structured approach not only confirms the hypothesis but also helps develop critical thinking and problem-solving skills.

Binomial Expansion

Binomial expansion is a technique used to expand expressions that involve powers of binomials (expressions with two terms). For example, the expression $(a + b)^2$ expands to $a^2 + 2ab + b^2$. This principle is part of the broader binomial theorem, which describes the expansion of $(a + b)^n$ for any integer $n$.
In the context of the exercise, recognizing the form $8t_n + 1 = (2n + 1)^2$ relied on identifying and the property of a binomial squared. After rewriting $8t_n + 1$ as $4n^2 + 4n + 1$, it matches the expanded form of $(2n + 1)^2$:

The $4n^2$ term with $(2n)^2$.
The $4n$ term with $2 imes 2n \times 1$.
The constant $1$ from $1^2$.

Understanding binomial expansion helps to quickly recognize patterns and verify results, providing a robust tool in algebraic transformations and proofs.

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