Section A Theory Solutions Exponential Equations And Geometric Progressions

1. Solving Exponential Equations and Geometric Progressions

1. a. Finding the value of x in the exponential equation

To find the value of x for which the equation $5^{2x} + 1 = 26(5^{x-1})$ holds true, we need to manipulate the equation and use substitution to simplify it. Exponential equations like this one often require a clever approach to solve. Let's start by rewriting the equation to make it more manageable.

First, rewrite the term $5^{x-1}$ as $5^x / 5$. The equation then becomes:

$5^{2x} + 1 = 26(5^x / 5)$

Multiply both sides of the equation by 5 to eliminate the fraction:

$5(5^{2x}) + 5 = 26(5^x)$

Now, let's rewrite $5^{2x}$ as $(5^x)^2$. This substitution will help us form a quadratic equation. The equation now looks like this:

$5(5^x)^2 + 5 = 26(5^x)$

Rearrange the terms to set the equation to zero:

$5(5^x)^2 - 26(5^x) + 5 = 0$

Now, let's introduce a substitution to simplify the equation further. Let $y = 5^x$. The equation transforms into a quadratic equation in terms of y:

$5y^2 - 26y + 5 = 0$

This quadratic equation can be factored. We are looking for two numbers that multiply to $5 * 5 = 25$ and add up to -26. These numbers are -25 and -1. So, we can rewrite the middle term as $-26y = -25y - y$:

$5y^2 - 25y - y + 5 = 0$

Now, factor by grouping:

$5y(y - 5) - 1(y - 5) = 0$

$(5y - 1)(y - 5) = 0$

This gives us two possible solutions for y:

$5y - 1 = 0$ or $y - 5 = 0$

Solving for y:

$y = 1/5$ or $y = 5$

Now, we substitute back $y = 5^x$ to find the values of x:

For $y = 1/5$:

$5^x = 1/5$

$5^x = 5^{-1}$

Therefore, $x = -1$

For $y = 5$:

$5^x = 5$

$5^x = 5^1$

Therefore, $x = 1$

Thus, the values of x that satisfy the equation are -1 and 1. This detailed solution demonstrates how to approach exponential equations by using substitution and factoring techniques to find the unknown values.

1. b. Finding the third term of a Geometric Progression

To determine the third term of a Geometric Progression (G.P.) given that the fourth and sixth terms are 54 and 486 respectively, we must first understand the fundamental properties of a G.P. In a geometric progression, each term is obtained by multiplying the previous term by a constant factor, known as the common ratio (r). Understanding this key concept allows us to set up equations and solve for the unknowns.

The general form of a term in a G.P. is given by:

$a_n = ar^{n-1}$

where:

• $a_n$ is the nth term,
• a is the first term,
• r is the common ratio, and
• n is the term number.

Given that the fourth term ($a_4$) is 54 and the sixth term ($a_6$) is 486, we can write the following equations:

$a_4 = ar^{4-1} = ar^3 = 54$ (Equation 1)

$a_6 = ar^{6-1} = ar^5 = 486$ (Equation 2)

To find the common ratio r, we can divide Equation 2 by Equation 1:

$(ar^5) / (ar^3) = 486 / 54$

$r^2 = 9$

Since it is given that r > 0, we take the positive square root:

$r = √9 = 3$

Now that we have the common ratio, we can substitute it back into Equation 1 to find the first term a:

$a(3^3) = 54$

$27a = 54$

$a = 54 / 27$

$a = 2$

Now that we have both the first term a = 2 and the common ratio r = 3, we can find the third term ($a_3$) using the general formula:

$a_3 = ar^{3-1} = ar^2$

$a_3 = 2(3^2)$

$a_3 = 2 * 9$

$a_3 = 18$

Therefore, the third term of the G.P. is 18. This step-by-step solution illustrates how to use the properties of geometric progressions to solve for unknown terms, emphasizing the importance of understanding the relationship between consecutive terms and the common ratio. The ability to work with geometric progressions is crucial in various mathematical contexts, including sequences, series, and financial calculations.

2. Discussion Category: Mathematics

This question seems to indicate a category for discussion, which is mathematics. It does not present a specific problem or question to be solved. Therefore, it is more of a classification than a problem that requires a solution. In an academic or educational context, categorizing a topic under mathematics is essential for organizing and structuring the subject matter. Mathematics as a broad field encompasses various sub-disciplines, including algebra, calculus, geometry, and number theory. Each of these branches deals with different aspects of mathematical principles and their applications.

To further elaborate on the significance of mathematics as a discussion category, it is important to consider the wide range of topics and concepts that fall under this umbrella. Mathematics is not merely about calculations; it is a way of thinking that involves logical reasoning, problem-solving, and abstract thinking. These skills are crucial not only in academic settings but also in real-world applications across various fields such as science, engineering, finance, and technology. The study of mathematics encourages critical thinking and analytical abilities, which are highly valued in today's job market. Understanding mathematical principles provides a foundation for advanced studies in various scientific and technical disciplines.

The beauty of mathematics lies in its precision and universality. Mathematical truths are consistent and applicable regardless of cultural or geographical context. This makes mathematics a powerful tool for communication and collaboration across different disciplines and cultures. For example, mathematical models are used in climate science to predict future climate patterns, in economics to analyze market trends, and in computer science to develop algorithms and software. The applications of mathematics are vast and continue to expand as new technologies and challenges emerge. In an educational setting, mathematics serves as a cornerstone of the curriculum, providing students with the essential skills needed to succeed in various aspects of life. From basic arithmetic to advanced calculus, mathematical concepts build upon each other, fostering a deep understanding of the world around us. The ability to reason mathematically and solve problems effectively is a lifelong asset, enabling individuals to make informed decisions and contribute to society in meaningful ways. The exploration of mathematics also involves a creative process, where new ideas and solutions are developed through abstract thinking and logical deduction. Mathematicians often work collaboratively, sharing insights and challenging each other's assumptions to push the boundaries of knowledge. This collaborative spirit fosters innovation and progress in the field of mathematics. Overall, categorizing a topic under mathematics highlights its fundamental importance and the vast scope of its applications. It serves as a gateway to exploring a world of logical reasoning, problem-solving, and abstract thinking, which are essential skills for success in the 21st century.