Sum Of N Terms Of The AP √3, √12, √27, √48 A Comprehensive Guide
Understanding Arithmetic Progressions (A.P.) and the Sum of Their Terms
When dealing with sequences in mathematics, one of the most fundamental concepts is the arithmetic progression (A.P.). An A.P. is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. Understanding arithmetic progressions is crucial not only in academic mathematics but also in various real-world applications, such as financial calculations, physics problems, and computer science algorithms. The sum of the terms in an A.P., especially the sum of the first n terms, is a common calculation with a well-established formula, making it easier to solve problems involving such sequences. Before we dive into the specific problem, let's solidify our understanding of the key concepts.
An arithmetic progression can be represented as a, a + d, a + 2d, a + 3d, and so on, where 'a' is the first term and 'd' is the common difference. The nth term of an A.P. can be found using the formula: an = a + (n - 1)d. The sum of the first n terms of an A.P., often denoted as Sn, is given by the formula: Sn = n/2 [2a + (n - 1)d]. This formula is derived by pairing the first and last terms, the second and second-to-last terms, and so on, each pair summing to the same value. This provides a straightforward method to calculate the sum without having to add each term individually, especially when n is large. Now that we have refreshed the basic concepts and formulas, we can apply them to the problem at hand. Identifying the first term and the common difference in the given sequence is the first step towards calculating the sum of its first n terms, and this initial analysis is crucial for choosing the correct approach.
Analyzing the Given Arithmetic Progression: √3, √12, √27, √48
To solve the problem, we first need to confirm that the given sequence √3, √12, √27, √48 is indeed an arithmetic progression. This involves checking if the difference between consecutive terms is constant. Let's analyze the terms: √12 - √3 = 2√3 - √3 = √3, √27 - √12 = 3√3 - 2√3 = √3, √48 - √27 = 4√3 - 3√3 = √3. Since the difference between consecutive terms is consistently √3, we can confirm that this is an arithmetic progression with a common difference (d) of √3. Identifying the common difference is a crucial step because it's a key parameter in the formula for the sum of an arithmetic progression. The first term (a) of the sequence is clearly √3. With both the first term and the common difference identified, we are now well-equipped to apply the formula for the sum of the first n terms.
Now, let's use the formula for the sum of the first n terms of an A.P.: Sn = n/2 [2a + (n - 1)d]. Substituting a = √3 and d = √3 into the formula, we get: Sn = n/2 [2√3 + (n - 1)√3]. Simplifying the expression inside the brackets, we have: Sn = n/2 [2√3 + n√3 - √3] = n/2 [√3 + n√3]. Factoring out √3, we get: Sn = n/2 [√3(1 + n)]. Further simplification gives us: Sn = n(n + 1)√3 / 2. Therefore, the sum of the first n terms of the given A.P. is n(n + 1)√3 / 2. This result matches one of the given options, allowing us to confidently select the correct answer. The process of verifying that the sequence is an A.P., identifying the first term and common difference, and correctly applying the sum formula is a methodical approach that can be applied to similar problems. Understanding the steps involved is as important as arriving at the correct answer, as it builds a solid foundation for solving more complex problems.
Calculating the Sum of n Terms: Step-by-Step Solution
To calculate the sum of n terms of the given arithmetic progression, we follow a step-by-step approach to ensure accuracy and clarity. The given sequence is √3, √12, √27, √48, ... First, we identify the first term (a), which is √3. Next, we determine the common difference (d) by subtracting a term from its subsequent term. We already established that d = √3. Now, we apply the formula for the sum of the first n terms of an A.P., which is given by: Sn = n/2 [2a + (n - 1)d]. Substituting the values of a and d, we get:
Sn = n/2 [2(√3) + (n - 1)(√3)]
Now, we simplify the expression inside the brackets:
Sn = n/2 [2√3 + n√3 - √3]
Combine the terms involving √3:
Sn = n/2 [√3 + n√3]
Factor out √3:
Sn = n/2 [√3(1 + n)]
Finally, we can rewrite the expression as:
Sn = n(n + 1)√3 / 2
This step-by-step solution clearly demonstrates how to use the formula and simplify the expression to arrive at the correct sum. Breaking down the problem into smaller steps makes it easier to understand and reduces the chance of errors. The final expression, Sn = n(n + 1)√3 / 2, represents the sum of the first n terms of the given arithmetic progression. This result can be used to calculate the sum for any number of terms n, making it a versatile and important formula in solving related problems. This thorough approach to problem-solving ensures a deep understanding of the underlying mathematical principles.
Identifying the Correct Option: n(n+1)√3 / 2
After calculating the sum of the n terms of the arithmetic progression √3, √12, √27, √48, we arrived at the expression: Sn = n(n + 1)√3 / 2. Now, we need to identify which of the given options matches our calculated result. The options provided are:
- n(n-1)√3 / 2
- n(n+1)√3 / 2
- n(n+1)√3
- √3n(n-1)
Comparing our result with the given options, it is clear that option (2), n(n+1)√3 / 2, is the correct match. This confirms that our calculations and simplifications were accurate. Identifying the correct option is the final step in solving the problem and requires careful attention to detail. By systematically working through the problem, from identifying the A.P. to applying the sum formula and simplifying the expression, we were able to confidently arrive at the correct answer. This process not only helps in solving this particular problem but also reinforces the understanding of arithmetic progressions and their properties. In mathematics, accuracy and attention to detail are paramount, and this exercise demonstrates the importance of both.
Conclusion: Mastering Arithmetic Progressions
In conclusion, the problem of finding the sum of n terms of the arithmetic progression √3, √12, √27, √48 highlights the importance of understanding fundamental concepts and applying them methodically. We began by recognizing the sequence as an arithmetic progression, which allowed us to identify the first term and the common difference. These values were then used in the formula for the sum of the first n terms of an A.P. Through careful simplification, we arrived at the expression Sn = n(n + 1)√3 / 2, which matched option (2). This exercise underscores the significance of a step-by-step approach in solving mathematical problems. Breaking down a complex problem into smaller, manageable parts not only makes it easier to understand but also reduces the likelihood of errors.
Furthermore, this problem reinforces the broader concept of arithmetic progressions, which are a fundamental topic in mathematics. The ability to identify an A.P., determine its key parameters, and apply the appropriate formulas is a valuable skill that extends beyond academic settings. Understanding these concepts can be applied to various real-world scenarios involving sequences and series. The methodical approach we employed in solving this problem serves as a template for tackling similar challenges. By consistently practicing and applying these techniques, one can develop a deeper understanding of mathematics and improve problem-solving abilities. Ultimately, mastering arithmetic progressions is not just about solving textbook problems; it's about developing a logical and analytical mindset that is applicable across many disciplines. The successful resolution of this problem demonstrates the power of combining theoretical knowledge with practical application.
