Unveiling Sequences: Terms, Formulas, And Inequalities
Hey math enthusiasts! Let's dive into the fascinating world of sequences. We'll be exploring how to find the nth term using a given formula, determining if a specific value belongs to a sequence, and understanding inequalities between consecutive terms. So, grab your pencils and let's get started! This topic is crucial for anyone looking to build a strong foundation in algebra and calculus. Understanding sequences is like having the keys to unlock patterns in mathematics.
(a) The nth Term of a Sequence: A Deep Dive
(i) Finding T_30: Applying the Formula
Alright, guys, let's kick things off by finding the 30th term of a sequence. The formula for the nth term, denoted as T_n, is given by T_n = (n+1)/n^2. This formula tells us how to calculate any term in the sequence based on its position, represented by 'n'. To find T_30, we simply need to substitute 'n' with 30 in the formula. Doing so, we get: T_30 = (30+1)/30^2 = 31/900. So, T_30 equals 31/900. This means that the 30th term in this specific sequence has a value of 31/900. Keep in mind that 'n' represents the position of the term in the sequence – the first term, the second term, the third term, and so on. Understanding this relationship between 'n' and the term value is absolutely essential. Let's make sure we grasp the concept of sequences – they're essentially lists of numbers that follow a specific pattern or rule. In this case, our rule is defined by the formula T_n = (n+1)/n^2. Understanding and applying such formulas is the foundation of many mathematical concepts. This is like understanding the recipe for a cake; it tells us exactly how much of each ingredient to use to get the desired result. The ability to calculate specific terms, like T_30, provides us with concrete values to analyze the sequence’s behavior, such as whether the terms increase, decrease, or remain constant as 'n' gets larger. This is a super important skill when you start looking at more complex mathematical concepts like limits and series. The use of this formula is a pretty straightforward process, but it's super important to avoid making silly mistakes during calculations. Pay close attention to the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right) – or PEMDAS/BODMAS! It might seem like a simple calculation, but it serves as a foundation for understanding more complex sequence problems. Remember, practice makes perfect, so keep practicing these calculations, and you'll become a pro in no time.
(ii) Why 29/785 Isn't a Term in the Sequence
Now, let's figure out if a certain value, 29/785, actually belongs to our sequence. We need to determine if there exists a whole number 'n' that satisfies the equation (n+1)/n^2 = 29/785. To do this, we'll cross-multiply to get rid of the fractions: 785(n+1) = 29n^2. This simplifies to 785n + 785 = 29n^2. Rearranging, we get a quadratic equation: 29n^2 - 785n - 785 = 0. We can solve this quadratic equation using the quadratic formula: n = (-b ± √(b^2 - 4ac)) / 2a, where a=29, b=-785, and c=-785. Solving the quadratic equation gives us a value of 'n'. However, if 'n' turns out to be a non-integer, or a negative value, it can't be a valid term in our sequence because 'n' must represent a positive whole number (the position of the term). In fact, solving that quadratic equation is going to result in a non-integer value for n. This means there's no whole number 'n' that satisfies the equation. Consequently, 29/785 is not a term in the sequence. To truly understand this, remember that 'n' must always be a positive integer because it represents the position of the term in the sequence. Therefore, you cannot have a term at position 2.5 or -3; it's always term number 1, 2, 3, and so on. Another way to prove this is not a term is by noticing that if (n+1)/n^2 = 29/785, then n+1 must be a factor of 29, and n^2 must be a factor of 785. 29 is a prime number and has only two factors, 1 and 29. The square root of 785 is not an integer. So it will not be possible to satisfy both conditions. This is a pretty common type of question that tests your understanding of the formula and what it means for a number to belong to a sequence. The key takeaway here is that you're looking for an integer solution for 'n'. If you don't find one, then the given value isn't a member of the sequence. Always check if your solution for 'n' makes sense in the context of the sequence. Does the calculated value align with the expected behavior of the sequence? This type of problem also emphasizes the importance of mathematical reasoning and critical thinking. It's not just about plugging numbers into a formula but understanding the underlying mathematical principles at play.
(iii) Comparing T_n and T_{n+1}: Unveiling the Inequality
Time to tackle inequalities! We want to determine whether T_n is greater than, less than, or equal to T_n+1}. In other words, we're comparing a term in the sequence with its next term. Let's start by writing out the formulas for T_n and T_{n+1} = ((n+1)+1)/(n+1)^2 = (n+2)/(n+1)^2. Now, let's compare them. We need to decide whether T_n > T_n+1}, T_n < T_{n+1}, or T_n = T_{n+1}. Let's examine the expression: (n+1)/n^2 - (n+2)/(n+1)^2. Finding a common denominator and simplifying, we get: [(n+1)^3 - n^2(n+2)] / [n2(n+1)2] = [n^3 + 3n^2 + 3n + 1 - n^3 - 2n^2] / [n2(n+1)2] = (n^2 + 3n + 1) / [n2(n+1)2]. Since 'n' is a positive integer, both the numerator (n^2 + 3n + 1) and the denominator [n2(n+1)2] are always positive. This means T_n - T_{n+1} is positive, which implies that T_n > T_{n+1}. Therefore, T_n > T_{n+1} for all positive integers n. This means that the terms in the sequence are always decreasing. This is an important analysis that demonstrates the monotonic behavior of the sequence. It's decreasing for all positive values of 'n'. Essentially, as 'n' gets larger (moving further along the sequence), the value of each term gets smaller. This is why the inequality is true. We're effectively proving that the sequence is strictly decreasing. This skill is critical for understanding the overall behavior of the sequence. Recognizing that T_n > T_{n+1} lets us conclude that this sequence is a decreasing sequence. This is a crucial concept when exploring convergence and divergence later in higher mathematics. This part of the question really gets at your conceptual understanding. It's not just about plugging in numbers; it’s about comparing the terms and reasoning about their relationship. It shows that you understand the big picture! A great way to check your work would be to pick a few values of 'n' like 1, 2, and 3, and calculate the terms and see if your answers for the inequality check out.
(b) The First Term of a Sequence: Exploring Properties
This section is not provided in the prompt. I am skipping this section. Understanding and analyzing sequences is super valuable in mathematics, providing insights into patterns and relationships between numbers. These kinds of problems are building blocks for more advanced topics like calculus. Don't be afraid to experiment with different values and analyze the results. Keep up the excellent work; you're doing great!