Sums, Exponential Equations, And Series Expansions In Mathematics
(2)(i) Summing Even Numbers Up to 2001
Even numbers play a crucial role in mathematics, possessing unique properties and appearing in various mathematical contexts. When we delve into the realm of number theory, one common task involves finding the sum of a series of numbers that follow a specific pattern. In this instance, we are tasked with determining the sum of all even numbers up to 2001. This seemingly simple problem unlocks a fascinating exploration of arithmetic progressions and the elegance of mathematical formulas. To effectively tackle this, we must first recognize that even numbers form an arithmetic progression (AP). An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. For even numbers, this common difference is consistently 2. The series we're considering starts with 2, proceeds through 4, 6, and so on, until it reaches 2000. Note that 2001 is an odd number, so the last even number before it is 2000.
To find the sum of an arithmetic progression, we can employ a well-established formula. The sum S of an AP is given by S = (n/2)(a + l), where n represents the number of terms in the series, a is the first term, and l is the last term. In our case, a is 2 and l is 2000. The challenge now lies in determining the value of n, which represents the number of even numbers from 2 to 2000. Since every other number is even, we can find n by dividing the last even number by 2. Thus, n = 2000 / 2 = 1000. Now that we have all the necessary components, we can substitute these values into the formula: S = (1000 / 2)(2 + 2000) = 500 * 2002 = 1001000. Therefore, the sum of all even numbers up to 2001 is 1,001,000. This result underscores the power of arithmetic progressions in simplifying what might initially appear to be a complex summation problem.
This calculation not only provides a numerical answer but also highlights the beauty of mathematical patterns. The ability to recognize and apply formulas like the sum of an arithmetic progression allows us to efficiently solve problems that would otherwise require tedious manual calculations. The concept of arithmetic progressions extends beyond just even numbers; it applies to any sequence where there is a constant difference between terms. Understanding and mastering these fundamental concepts is crucial for progressing in mathematics and tackling more advanced topics. Furthermore, the process of solving this problem reinforces the importance of breaking down a problem into smaller, manageable parts. By identifying the arithmetic progression, determining the relevant parameters (a, l, and n), and applying the correct formula, we arrive at the solution in a clear and concise manner. This methodical approach is a valuable skill in mathematics and in problem-solving in general.
(2)(b) Solving the Exponential Equation $3^{2t-1} = 7^{t+2}$ for $t$
Exponential equations are fundamental in mathematics and find applications in various fields, including physics, engineering, and finance. These equations involve variables in the exponents, and solving them often requires the use of logarithms. The given equation, $3^{2t-1} = 7^{t+2}$, is a classic example of an exponential equation. To solve for t, we need to employ logarithmic properties to bring the exponents down and isolate t. The key principle here is that if $a^x = b^y$, then taking the logarithm of both sides with the same base will allow us to equate the exponents (with some modifications). The most common bases for logarithms are base 10 (common logarithm) and base e (natural logarithm, denoted as ln). Either base can be used to solve this equation, but for the sake of demonstration, let's use the natural logarithm (ln).
Applying the natural logarithm to both sides of the equation, we get: ln($3^2t-1}$) = ln($7^{t+2}$). A crucial property of logarithms states that ln($a^b$) = b ln(a). Applying this property, we can rewrite the equation as(3) = (t + 2) extln}(7)$. Now, we have a linear equation in terms of t. The next step is to expand both sides and rearrange the terms to isolate t. Expanding the equation, we get(3) - extln}(3) = t ext{ln}(7) + 2 ext{ln}(7)$. Now, let's group the terms containing t on one side and the constant terms on the other side(3) - t extln}(7) = 2 ext{ln}(7) + ext{ln}(3)$. Factor out t from the left side(3) - extln}(7)) = 2 ext{ln}(7) + ext{ln}(3)$. Finally, to solve for t, divide both sides by the coefficient of t(7) + extln}(3)}{2 ext{ln}(3) - ext{ln}(7)}$. This is the exact solution for t. To obtain a numerical approximation, we can use a calculator to evaluate the logarithms. ln(7) β 1.9459 and ln(3) β 1.0986. Substituting these values{2(1.0986) - 1.9459} β rac{3.8918 + 1.0986}{2.1972 - 1.9459} β rac{4.9904}{0.2513} β 19.857$. Therefore, the approximate solution for t is 19.857.
This problem demonstrates the importance of understanding logarithmic properties in solving exponential equations. The ability to manipulate logarithms allows us to transform complex equations into simpler, solvable forms. The process of taking logarithms, expanding terms, and isolating the variable is a common technique in mathematics and is essential for solving a wide range of problems. Moreover, this example highlights the power of logarithms in handling equations where the variable appears in the exponent. Without logarithms, solving such equations would be significantly more challenging. The solution also emphasizes the distinction between an exact solution (in terms of logarithms) and a numerical approximation, which is often necessary for practical applications. The ability to obtain both types of solutions is a valuable skill in mathematics and science. Furthermore, understanding the properties of logarithms is crucial for understanding many scientific and engineering principles, making this a fundamental concept in STEM fields.
(3)(a) Finding the First 4 Terms of the Expansion of $ rac{1}{(1+x2)4}$
Series expansions are a powerful tool in mathematics, allowing us to represent complex functions as infinite sums of simpler terms. The binomial theorem and its generalizations provide a systematic way to expand expressions of the form $(a + b)^n$, where n can be a positive integer, a negative integer, or a fraction. In this problem, we are tasked with finding the first 4 terms of the expansion of $ rac{1}{(1+x2)4}$, which can be rewritten as $(1 + x2){-4}$. This falls under the category of a binomial series expansion, where the exponent is a negative integer. The binomial theorem for negative exponents states that for any real number n and |x| < 1:
(1 + x)^n = 1 + nx + rac{n(n-1)}{2!}x^2 + rac{n(n-1)(n-2)}{3!}x^3 + ...
In our case, we have $(1 + x2){-4}$, so we can substitute n = -4 and replace x with $x^2$ in the general formula. Let's calculate the first four terms:
- Term 1: 1
- Term 2: nx = (-4)($x^2$) = -4$x^2$
- Term 3: $ rac{n(n-1)}{2!}x^2$ = $ rac{(-4)(-4-1)}{2!}(x2)2$ = $ rac{(-4)(-5)}{2}x^4$ = 10$x^4$
- Term 4: $ rac{n(n-1)(n-2)}{3!}x^3$ = $ rac{(-4)(-4-1)(-4-2)}{3!}(x2)3$ = $ rac{(-4)(-5)(-6)}{6}x^6$ = -20$x^6$
Therefore, the first 4 terms of the expansion of $ rac1}{(1+x2)4}$ are{(1+x2)4}$ near x = 0. The more terms we include in the expansion, the better the approximation becomes.
The process of finding the series expansion involves applying the binomial theorem and carefully calculating the coefficients. It also highlights the importance of understanding the conditions under which the expansion is valid (in this case, |x| < 1). The binomial series has numerous applications in mathematics, physics, and engineering. For instance, it can be used to approximate solutions to equations, calculate probabilities, and model physical phenomena. Understanding and mastering the binomial series expansion is a valuable skill for anyone working in these fields. Moreover, this example underscores the connection between different areas of mathematics, such as algebra and calculus. The binomial theorem is an algebraic result, but its applications often extend into calculus, particularly in the study of power series and Taylor series.
(3)(b) Analyzing Arithmetic Progression (AP) with First Term $b$ and Common Difference Unknown
Arithmetic Progressions (APs) are fundamental sequences in mathematics, characterized by a constant difference between consecutive terms. They appear in various mathematical contexts and have numerous applications in fields like physics, engineering, and finance. An arithmetic progression is defined by its first term, often denoted as a, and its common difference, d. The nth term of an AP, denoted as $a_n$, can be expressed as $a_n = a + (n - 1)d$. The sum of the first n terms of an AP, denoted as $S_n$, is given by $S_n = rac{n}{2}[2a + (n - 1)d]$ or $S_n = rac{n}{2}(a + l)$, where l is the last term. In this problem, we are given an AP with the first term b, but the common difference is not explicitly provided. This means we need to analyze the given information and potentially derive relationships or solve for unknowns based on additional conditions or terms provided.
Without further information, we can express the general terms of the AP in terms of b and d. The second term would be b + d, the third term b + 2d, the fourth term b + 3d, and so on. To fully analyze the AP, we would typically need more information, such as the value of the common difference d, the value of a specific term in the sequence, or a relationship between the terms. For example, if we were given the second term, we could easily solve for d. Similarly, if we were given the sum of the first n terms, we could set up an equation involving b, d, and n, and potentially solve for one or more of these variables.
The analysis of arithmetic progressions often involves solving problems related to finding specific terms, determining the sum of a certain number of terms, or identifying the common difference. These problems can range from straightforward applications of the formulas to more complex scenarios that require algebraic manipulation and problem-solving skills. Understanding the properties of arithmetic progressions is crucial for tackling these types of problems. The concept of arithmetic progressions also extends to more advanced mathematical topics, such as series and sequences in calculus. In calculus, understanding sequences and series is essential for defining concepts like limits, continuity, and convergence.
Furthermore, the applications of arithmetic progressions are not limited to theoretical mathematics. They also appear in practical contexts, such as modeling linear growth or decay, calculating simple interest, and analyzing patterns in data. For example, if a person saves a fixed amount of money each month, the total savings over time will form an arithmetic progression. Similarly, the depreciation of an asset at a constant rate can be modeled using an arithmetic progression. Therefore, the study of arithmetic progressions is not only important for mathematical understanding but also for real-world applications. To further illustrate, consider a scenario where we are given that the third term of the AP is 10 and the fifth term is 16. We can set up a system of equations to solve for b and d. The third term is b + 2d = 10, and the fifth term is b + 4d = 16. Subtracting the first equation from the second, we get 2d = 6, so d = 3. Substituting d = 3 into the first equation, we get b + 2(3) = 10, so b = 4. Thus, with additional information, we can fully define the arithmetic progression.
