Solve The Equation 5/14 = N(n-1) / (8 * 7) For N.

Embark on a mathematical journey as we dissect the equation $\frac{5}{14} = \frac{n(n-1)}{8 \cdot 7}$, delving into the intricacies of its structure and unraveling the value of 'n' that satisfies this elegant expression. This exploration transcends the mere act of solving an equation; it's a voyage into the heart of mathematical reasoning, where we'll employ a symphony of algebraic techniques and logical deductions to arrive at a solution that not only answers the question but also illuminates the beauty of mathematical problem-solving.

Dissecting the Equation: A Symphony of Fractions and Quadratics

At first glance, the equation $\frac{5}{14} = \frac{n(n-1)}{8 \cdot 7}$ presents itself as a harmonious blend of fractions and algebraic expressions. To truly grasp its essence, we must embark on a journey of dissection, meticulously examining each component and its interplay with the others. On one side, we encounter the fraction $\frac{5}{14}$, a numerical entity that embodies the concept of proportion, representing five parts out of a whole divided into fourteen equal segments. On the other side, we find the algebraic expression $\frac{n(n-1)}{8 \cdot 7}$, a composition of variables and arithmetic operations that hints at the potential for quadratic behavior. The variable 'n' takes center stage, engaging in a dance of multiplication and subtraction, its value yet to be unveiled. The denominator, $8 \cdot 7$, stands as a numerical anchor, providing a sense of scale and grounding the expression in the realm of arithmetic.

The equation, like a bridge connecting two distinct mathematical shores, asserts an equality between these seemingly disparate entities. It proclaims that the fraction $\frac{5}{14}$ and the algebraic expression $\frac{n(n-1)}{8 \cdot 7}$ are but two faces of the same mathematical coin, their values intrinsically intertwined. To solve this equation is to embark on a quest to uncover the value of 'n' that renders this proclamation true, to find the numerical key that unlocks the hidden harmony within the equation.

The journey begins with a careful simplification of the equation, a process akin to clearing away the underbrush to reveal the underlying structure. We observe that the denominator on the right-hand side, $8 \cdot 7$, can be simplified to 56, transforming the equation into $\frac{5}{14} = \frac{n(n-1)}{56}$. This seemingly simple act of arithmetic simplification sets the stage for the next act in our mathematical drama, where we shall employ the powerful technique of cross-multiplication to further unravel the mysteries of the equation.

Cross-Multiplication: A Bridge Across the Divide

Cross-multiplication, a venerable technique in the mathematician's toolkit, stands as a bridge across the divide that separates the two sides of an equation involving fractions. It is a powerful maneuver that allows us to transform a proportion into a more manageable form, paving the way for the application of algebraic techniques. In the context of our equation, $\frac{5}{14} = \frac{n(n-1)}{56}$, cross-multiplication beckons as a promising path towards a solution.

To execute this maneuver, we embark on a carefully choreographed dance of multiplication, where the numerator of one fraction is multiplied by the denominator of the other, and vice versa. The result is a new equation, one that no longer harbors the fractional form, but instead presents itself as a linear equation, ripe for algebraic manipulation.

In our case, cross-multiplication dictates that we multiply 5 by 56, and 14 by $n(n-1)$, yielding the equation $5 \cdot 56 = 14 \cdot n(n-1)$. This seemingly simple act of multiplication has profound consequences, for it transforms the equation from a realm of fractions to a realm of algebraic expressions, where the familiar tools of algebra can be brought to bear.

With the equation now in a more amenable form, we proceed to simplify, reducing the numerical coefficients and paving the way for the isolation of the variable 'n'. We observe that $5 \cdot 56$ equals 280, and $14 \cdot n(n-1)$ can be expanded to $14n^2 - 14n$, transforming the equation into $280 = 14n^2 - 14n$. This equation, though seemingly more complex than the original, holds the key to unlocking the value of 'n', for it is a quadratic equation, a type of equation that mathematicians have studied for centuries, and for which a rich arsenal of solution techniques exists.

The Quadratic Equation: Unveiling the Roots

The equation $280 = 14n^2 - 14n$, born from the ashes of cross-multiplication, now stands before us as a quadratic equation, a mathematical entity of great significance and power. Quadratic equations, characterized by the presence of a squared term, arise in a multitude of contexts, from the trajectory of projectiles to the optimization of economic models. To solve a quadratic equation is to find the values of the variable that satisfy the equation, the roots that lie hidden within its algebraic structure.

To tackle this quadratic equation, we first embark on a journey of simplification, seeking to reduce the numerical coefficients and bring the equation into a more manageable form. We observe that all the terms in the equation are divisible by 14, suggesting a path towards simplification. Dividing both sides of the equation by 14, we arrive at the more elegant form $20 = n^2 - n$, an equation that retains the essence of the original while presenting a cleaner facade.

To further refine the equation, we rearrange the terms, bringing all terms to one side and setting the equation equal to zero. This maneuver transforms the equation into the standard quadratic form, $n^2 - n - 20 = 0$, a form that is particularly amenable to solution by factoring.

Factoring, a venerable technique in the mathematician's toolkit, involves expressing a quadratic expression as the product of two linear expressions. In the case of $n^2 - n - 20$, we seek two numbers that multiply to -20 and add up to -1. After a moment's reflection, we realize that -5 and 4 satisfy these criteria, allowing us to factor the quadratic expression as $(n - 5)(n + 4) = 0$.

The factored equation, $(n - 5)(n + 4) = 0$, presents a profound insight into the solutions of the quadratic equation. It proclaims that the product of two expressions is zero, implying that at least one of the expressions must be zero. Thus, either $n - 5 = 0$ or $n + 4 = 0$. Solving these linear equations, we arrive at the solutions $n = 5$ and $n = -4$, the roots of the quadratic equation, the values of 'n' that satisfy the equation.

Validation and Conclusion: The Triumph of Reason

With the candidate solutions in hand, we embark on the final act of our mathematical drama: validation. To ensure the correctness of our solutions, we must subject them to the crucible of the original equation, $\frac{5}{14} = \frac{n(n-1)}{8 \cdot 7}$, verifying that they indeed satisfy the equality.

First, we consider the solution $n = 5$. Substituting this value into the equation, we obtain $\frac{5}{14} = \frac{5(5-1)}{8 \cdot 7}$, which simplifies to $\frac{5}{14} = \frac{20}{56}$. Further simplification reveals that $\frac{20}{56}$ is indeed equal to $\frac{5}{14}$, confirming that $n = 5$ is a valid solution.

Next, we turn our attention to the solution $n = -4$. Substituting this value into the equation, we obtain $\frac{5}{14} = \frac{-4(-4-1)}{8 \cdot 7}$, which simplifies to $\frac{5}{14} = \frac{20}{56}$. Again, simplification reveals that $\frac{20}{56}$ is equal to $\frac{5}{14}$, confirming that $n = -4$ is also a valid solution.

Thus, our journey through the equation $\frac{5}{14} = \frac{n(n-1)}{8 \cdot 7}$ culminates in a triumphant conclusion. We have successfully unveiled the two values of 'n' that satisfy the equation: $n = 5$ and $n = -4$. These solutions stand as a testament to the power of mathematical reasoning, the ability of algebraic techniques to unravel the mysteries hidden within equations.

This exploration has been more than just a solution to a mathematical problem; it has been a journey into the heart of mathematics itself, a voyage that has revealed the beauty of equations, the elegance of algebraic techniques, and the satisfaction of arriving at a solution through the power of reason. The equation $\frac{5}{14} = \frac{n(n-1)}{8 \cdot 7}$, once a puzzle, now stands as a symbol of our mathematical triumph.

Conclusion

In conclusion, the equation $\frac{5}{14} = \frac{n(n-1)}{8 \cdot 7}$ has been thoroughly dissected and solved, revealing the values of 'n' to be 5 and -4. This journey has showcased the power of mathematical techniques such as cross-multiplication, simplification, and factoring, highlighting the interconnectedness of mathematical concepts. The successful solution serves as a testament to the beauty and elegance inherent in mathematical problem-solving.