Finding N In The Expansion Of (5 + Nx)^2 (1 + (3/5)x)^n

Introduction

In this article, we delve into the fascinating realm of binomial expansions to solve a specific problem. Our primary objective is to determine the value of n given the equation extbf{equation (5 + nx)^2 (1 + (3/5)x)^n = 25 + 100x + ⋯}. This involves a meticulous analysis of the binomial theorem, expansion techniques, and careful comparison of coefficients. Understanding binomial expansions is not only crucial for solving mathematical problems but also has significant applications in various fields such as statistics, physics, and computer science. This problem particularly highlights the importance of algebraic manipulation and the ability to extract relevant information from complex expressions. In this detailed exploration, we will break down the given equation, expand each part, and then equate the coefficients to find the unknown value of n. We will also touch upon the underlying principles and theorems that make this process possible, ensuring a comprehensive understanding of the solution.

Understanding the Problem

Before diving into the solution, it’s essential to fully grasp the problem statement. We are presented with an equation involving the product of two expressions: extbf{(5 + nx)^2} and extbf{(1 + (3/5)x)^n}. The result of this product is given in the form of a power series, specifically 25 + 100x + ⋯, where the ellipsis indicates that there are higher-order terms that we don't need to consider for this problem. The core task is to find the value of n. This suggests that we need to expand both expressions and then compare the coefficients of the terms on both sides of the equation. The coefficient of the x term is particularly significant in this case, as it provides a direct link to n. We will leverage the binomial theorem to expand extbf{(1 + (3/5)x)^n} and basic algebraic expansion for extbf{(5 + nx)^2}. By carefully equating the coefficients, we will establish a relationship that allows us to solve for n. The ability to accurately expand these expressions and equate coefficients is a fundamental skill in algebra and calculus, making this problem an excellent exercise in mathematical reasoning.

Expanding (5 + nx)^2

Let's begin by expanding the first part of the expression, extbf{(5 + nx)^2}. This can be done using the standard algebraic identity extbf{(a + b)^2 = a^2 + 2ab + b^2}. Applying this identity, we get:

extbf{(5 + nx)^2 = 5^2 + 2 * 5 * nx + (nx)^2}

Simplifying this gives:

extbf{(5 + nx)^2 = 25 + 10nx + n^2x^2}

This expansion is straightforward and provides us with a quadratic expression in terms of x. The coefficients of the terms are crucial for our next steps. Specifically, we have the constant term as 25, the coefficient of x as 10n, and the coefficient of x^2 as n^2. These coefficients will play a significant role when we multiply this expansion with the expansion of the second part of the expression. The simplicity of this expansion contrasts with the binomial expansion required for the second term, but both are essential to solving the problem. By accurately expanding this quadratic expression, we've laid a solid foundation for the subsequent steps.

Expanding (1 + (3/5)x)^n using the Binomial Theorem

Now, let's tackle the second part of the expression: extbf{(1 + (3/5)x)^n}. This requires the use of the binomial theorem, which states that for any positive integer n:

extbf{(1 + x)^n = 1 + nx + (n(n-1) / 2!)x^2 + (n(n-1)(n-2) / 3!)x^3 + ⋯}

In our case, we have extbf{(1 + (3/5)x)^n}, so we need to substitute (3/5)x for x in the binomial expansion. We are primarily interested in the terms up to the first power of x because the given equation only provides us with terms up to 100x. Thus, we can write:

extbf{(1 + (3/5)x)^n ≈ 1 + n(3/5)x + ⋯}

This expansion gives us the first two terms of the series. The constant term is 1, and the coefficient of x is (3n/5). These values are crucial when we multiply this expansion with the expansion of extbf{(5 + nx)^2}. The binomial theorem is a powerful tool in algebra and calculus, allowing us to expand expressions of the form extbf{(a + b)^n} for any positive integer n. In this context, it provides us with the necessary framework to understand and solve the problem at hand. By focusing on the initial terms of the expansion, we simplify the problem and make it more manageable.

Multiplying the Expansions

Now that we have expanded both parts of the expression, we need to multiply them together. We have:

extbf{(5 + nx)^2 = 25 + 10nx + n^2x^2}

and

extbf{(1 + (3/5)x)^n ≈ 1 + (3n/5)x}

Multiplying these two expansions, we primarily focus on terms that result in a constant or x term, as these are the terms given in the problem. The multiplication is as follows:

extbf{(25 + 10nx + n^2x^2)(1 + (3n/5)x) = 25(1) + 25(3n/5)x + 10nx(1) + ⋯}

Simplifying this, we get:

extbf{25 + 15nx + 10nx + ⋯ = 25 + 25nx + ⋯}

Here, we've retained only the constant and the x terms, ignoring higher-order terms as they are not relevant to our problem. The coefficient of x in the resulting expansion is 25n. This is a critical result, as it directly relates to the value of n that we are trying to find. The process of multiplying these expansions highlights the importance of careful algebraic manipulation and attention to detail. By focusing on the relevant terms, we can simplify the problem and extract the necessary information to solve for n.

Equating Coefficients and Solving for n

We are given that the expansion of extbf{(5 + nx)^2 (1 + (3/5)x)^n} is extbf{25 + 100x + ⋯}. Comparing this with our expanded form extbf{25 + 25nx + ⋯}, we can equate the coefficients of the x terms:

extbf{25n = 100}

Now, we solve for n:

extbf{n = 100 / 25}

extbf{n = 4}

Therefore, the value of n is 4. This result is obtained by carefully equating the coefficients of the x terms in the expanded form and the given power series. The ability to equate coefficients is a fundamental technique in algebra and calculus, allowing us to solve for unknown variables in polynomial and power series expansions. In this case, it provides a direct and straightforward method to find the value of n. The solution highlights the importance of accurate algebraic manipulation and a clear understanding of the binomial theorem.

Conclusion

In conclusion, by expanding extbf{(5 + nx)^2} and extbf{(1 + (3/5)x)^n} using algebraic identities and the binomial theorem, respectively, and then equating the coefficients of the x terms, we found that the value of n is 4. This problem underscores the significance of binomial expansions and coefficient comparison in solving algebraic problems. The process involved breaking down the given expression, expanding each part accurately, and then carefully equating the coefficients to solve for the unknown variable. This approach is not only applicable to this specific problem but also provides a general framework for tackling similar challenges in algebra and calculus. The solution demonstrates the power of mathematical techniques in simplifying complex expressions and extracting meaningful information. Understanding these techniques is essential for further studies in mathematics and its applications in various scientific and engineering fields.