Sequence And Series Problems: Formulas And Calculations
Hey everyone! Let's dive into some cool math problems focusing on sequences and series. We'll break down how to find the formula for the nth term and how to calculate specific terms in a sequence. So, grab your thinking caps, and let's get started!
1. Finding the Formula for the nth Term
Let's kick things off with the first question: What's the formula for the nth term of the sequence 2, 6, 10...? This is a classic problem, and figuring it out involves spotting the pattern and translating that into an algebraic expression. When dealing with arithmetic sequences, the key is to identify the common difference between consecutive terms. In this sequence, the difference between each term is 4 (6 - 2 = 4, 10 - 6 = 4). This tells us we're dealing with a linear relationship, something of the form Un = an + b, where 'a' is the common difference and 'b' is some constant we need to figure out.
Now, let's delve deeper into how we can find the formula for the nth term. We know the general form for an arithmetic sequence is _Un = a_n + b*, where Un represents the nth term, 'a' is the common difference, 'n' is the term number, and 'b' is a constant we need to determine. From the sequence 2, 6, 10..., we've already identified that the common difference (a) is 4. So, our formula looks like Un = 4n + b. To find 'b', we can plug in the values from the first term of the sequence. When n = 1, Un = 2. So, 2 = 4(1) + b. Solving for 'b', we get b = 2 - 4 = -2. Therefore, the formula for the nth term of this sequence is Un = 4n - 2. Let's verify this formula with the given options. Option (a) Un = 2n doesn't fit because when n = 1, Un should be 2, but it doesn't account for the increasing difference. Option (b) Un = 2n + 2 gives Un = 4 when n = 1, which is incorrect. Option (c) Un = 4n - 1 is close, but when n = 1, it gives Un = 3, not 2. Option (d) Un = 4n + 1 gives Un = 5 when n = 1, which is also incorrect. However, option (e) Un = 4n - 2 correctly predicts Un = 2 when n = 1, Un = 6 when n = 2 (42 - 2 = 6), and Un = 10 when n = 3 (43 - 2 = 10). Thus, the correct formula is indeed Un = 4n - 2.
To further solidify our understanding, let’s consider another example. Suppose we have the sequence 3, 8, 13, 18.... The common difference here is 5. So, the formula starts as Un = 5n + b. When n = 1, Un = 3. Thus, 3 = 5(1) + b, which gives us b = 3 - 5 = -2. The formula for this sequence is Un = 5n - 2. We can check this by plugging in n = 2, which gives Un = 5(2) - 2 = 8, which matches the second term in the sequence. This methodical approach of identifying the common difference and solving for the constant term ensures we can accurately find the nth term formula for any arithmetic sequence. This skill is fundamental in more advanced mathematics, such as calculus and discrete mathematics, where sequences and series play a crucial role. Understanding how sequences behave allows us to model real-world phenomena and solve complex problems. For instance, in finance, sequences can be used to model the growth of investments over time, and in physics, they can describe the motion of objects under certain conditions. So, mastering this foundational concept is a valuable step in your mathematical journey.
2. Calculating a Specific Term in a Sequence
Next up, we're tackling the second question: What is the 41st term of the sequence 1, 4, 7, 10, 13...? Again, we see an arithmetic sequence, but this time we're asked to find a specific term rather than the general formula. The approach is similar: identify the common difference, write the formula, and then plug in the term number we're interested in. In this sequence, the common difference is 3. Using the same formula structure, Un = an + b, we get Un = 3n + b. The first term is 1, so when n = 1, Un = 1. Plugging these values in, 1 = 3(1) + b, which gives us b = -2. So, the formula for this sequence is Un = 3n - 2.
Now, let's dive into how we calculate the 41st term of the sequence 1, 4, 7, 10, 13... We've already established that the formula for the nth term of this arithmetic sequence is Un = 3n - 2. To find the 41st term, we simply need to substitute n = 41 into the formula. So, U41 = 3(41) - 2. Now, let’s do the math. 3 multiplied by 41 is 123. Then, we subtract 2 from 123, which gives us 121. Therefore, the 41st term of the sequence is 121. Looking at the options provided, we can see that option (b) 121 matches our calculation. To further illustrate the process, let's consider a similar problem. Suppose we want to find the 50th term of the sequence 2, 5, 8, 11... First, we identify the common difference, which is 3. The general formula is Un = 3n + b. For the first term, when n = 1, Un = 2. Plugging these values into the formula, we get 2 = 3(1) + b, which means b = -1. Thus, the formula for this sequence is Un = 3n - 1. To find the 50th term, we substitute n = 50 into the formula: U50 = 3(50) - 1. This simplifies to U50 = 150 - 1, which equals 149. So, the 50th term of the sequence is 149. This straightforward method of substituting the term number into the formula allows us to quickly find any term in an arithmetic sequence. The ability to calculate specific terms is crucial in various applications, such as predicting future values based on a known pattern. For example, in financial planning, we might use sequences to project the growth of savings or investments over time. In computer science, sequences are used in algorithms and data structures. Understanding how to find specific terms helps in solving practical problems and building a solid foundation in mathematics.
To find the 41st term, we substitute n = 41 into the formula: U41 = 3(41) - 2 = 123 - 2 = 121. So, the correct answer is b. 121. This type of question highlights the importance of accurately applying the formula and performing the arithmetic correctly.
3. Mathematics Discussion Category
Finally, let's categorize this discussion. It clearly falls under mathematics, specifically dealing with sequences and series. These topics are fundamental in algebra and calculus, and understanding them is crucial for more advanced mathematical studies.
Wrapping up, we've tackled some sequence problems, focusing on finding formulas and calculating specific terms. Remember, the key is to identify patterns, apply the formulas correctly, and double-check your work. Keep practicing, and you'll become a sequence and series pro in no time! If you guys have any questions or want to explore more problems, just let me know. Happy problem-solving!
