Finding The Sum Of First 13 Terms Of An Arithmetic Progression

by Admin 63 views

This article delves into the fascinating world of arithmetic progressions, focusing on a specific problem: determining the sum of the first 13 terms of an arithmetic progression given a particular condition. We will explore the fundamental concepts of arithmetic progressions, derive the necessary formulas, and apply them to solve the problem at hand. This comprehensive guide aims to provide a clear and concise understanding of the topic, making it accessible to learners of all levels.

H2: Understanding Arithmetic Progressions

At the heart of this problem lies the concept of an arithmetic progression. An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference, often denoted by 'd'. The first term of the progression is typically represented by 'a_1'. Therefore, an arithmetic progression can be written as:

a_1, a_1 + d, a_1 + 2d, a_1 + 3d, ...

Each term in the sequence can be expressed using the general formula:

a_n = a_1 + (n - 1)d

where 'a_n' represents the nth term of the progression. This formula is crucial for understanding and manipulating arithmetic progressions. In essence, it states that any term in the sequence can be found by adding the common difference (d) to the first term (a_1) a certain number of times (n-1). Understanding this fundamental principle is key to solving problems involving arithmetic progressions. For instance, if you know the first term and the common difference, you can easily calculate any term in the sequence. Conversely, if you know two terms in the sequence, you can determine the common difference and the first term. Let's consider some practical examples to solidify this understanding. Imagine a sequence where the first term is 2 and the common difference is 3. The sequence would then be 2, 5, 8, 11, and so on. Using the formula, we can find the 10th term: a_10 = 2 + (10 - 1) * 3 = 29. This simple calculation demonstrates the power of the general formula. Now, let's consider a slightly more complex scenario. Suppose we know that the 5th term of an arithmetic progression is 15 and the 9th term is 27. How would we find the first term and the common difference? We can set up two equations using the general formula: 15 = a_1 + 4d and 27 = a_1 + 8d. Solving this system of equations will give us the values of a_1 and d. This ability to manipulate the general formula and solve for unknowns is essential for tackling various arithmetic progression problems. The general formula acts as a cornerstone in the study of arithmetic progressions, providing a systematic way to analyze and predict the behavior of these sequences.

H2: The Sum of an Arithmetic Progression

Another fundamental concept related to arithmetic progressions is the sum of its terms. The sum of the first 'n' terms of an arithmetic progression, denoted by S_n, can be calculated using the following formula:

S_n = (n/2) * [2a_1 + (n - 1)d]

Alternatively, if the first term (a_1) and the last term (a_n) are known, the sum can also be calculated as:

S_n = (n/2) * (a_1 + a_n)

These formulas provide efficient ways to calculate the sum of an arithmetic progression without having to add up all the individual terms. The first formula is particularly useful when you know the first term (a_1), the common difference (d), and the number of terms (n). It allows you to directly compute the sum without needing to calculate each term individually. The second formula is advantageous when you know the first term (a_1) and the last term (a_n), as it simplifies the calculation by directly using these values. Both formulas are derived from the same underlying principle: the average of the first and last term, multiplied by the number of terms. To truly appreciate the elegance and practicality of these formulas, let's consider a few examples. Imagine an arithmetic progression with a first term of 3, a common difference of 2, and we want to find the sum of the first 10 terms. Using the first formula, we have: S_10 = (10/2) * [2*3 + (10 - 1)*2] = 5 * [6 + 18] = 5 * 24 = 120. This calculation demonstrates how quickly the sum can be determined using the formula. Now, let's consider a scenario where we know the first term is 5, the last term is 45, and we want to find the sum of the first 21 terms. Using the second formula, we have: S_21 = (21/2) * (5 + 45) = 10.5 * 50 = 525. This example highlights the convenience of the second formula when the last term is known. Understanding these formulas is paramount for solving various problems involving arithmetic progressions. They not only provide a shortcut for calculating the sum but also offer valuable insights into the relationship between the terms and the overall sum of the sequence. Mastering these formulas will empower you to tackle a wide range of arithmetic progression problems with confidence and efficiency.

H2: Problem Statement: Sum of the First 13 Terms

Now, let's address the specific problem at hand. We are tasked with finding the sum of the first 13 terms of an arithmetic progression (a_n), given that a_5 + a_9 = 16. This problem requires us to combine our understanding of arithmetic progressions and their sums to arrive at a solution. The key to solving this problem lies in utilizing the general formula for the nth term and the formula for the sum of an arithmetic progression. We need to express the given condition (a_5 + a_9 = 16) in terms of a_1 and d, and then use this information to find the sum of the first 13 terms. This approach involves a systematic application of the formulas and a bit of algebraic manipulation. Let's break down the problem step by step to ensure a clear understanding of the solution process. First, we need to express a_5 and a_9 using the general formula for the nth term. This will allow us to relate them to the first term (a_1) and the common difference (d). Then, we can substitute these expressions into the given condition (a_5 + a_9 = 16) and obtain an equation involving a_1 and d. This equation will provide us with a crucial relationship between the first term and the common difference. Next, we need to use this relationship to find the sum of the first 13 terms. We can use the formula for the sum of an arithmetic progression, which involves a_1, d, and the number of terms (n). By substituting the known values and the relationship we derived earlier, we can calculate the sum. This process may seem daunting at first, but by breaking it down into smaller, manageable steps, we can approach the problem with confidence. The key is to remember the fundamental formulas and apply them strategically. Each step builds upon the previous one, leading us closer to the final solution. This problem serves as an excellent example of how theoretical concepts can be applied to solve practical problems. It reinforces the importance of understanding the underlying principles of arithmetic progressions and their applications.

H2: Solution: Step-by-Step Calculation

  1. Express a_5 and a_9 using the general formula:

    • a_5 = a_1 + 4d
    • a_9 = a_1 + 8d

    This step involves applying the fundamental formula for the nth term of an arithmetic progression. By substituting n = 5 and n = 9 into the formula a_n = a_1 + (n - 1)d, we express the 5th and 9th terms in terms of the first term (a_1) and the common difference (d). This is a crucial step as it allows us to translate the given condition (a_5 + a_9 = 16) into an equation involving a_1 and d. The ability to manipulate the general formula and express terms in different forms is essential for solving arithmetic progression problems. This step highlights the importance of understanding the underlying structure of arithmetic progressions and how the terms relate to each other. By expressing a_5 and a_9 in this way, we are setting the stage for the next step, where we will use the given condition to establish a relationship between a_1 and d.

  2. Substitute these expressions into the given condition a_5 + a_9 = 16:

    (a_1 + 4d) + (a_1 + 8d) = 16

    This step involves substituting the expressions we derived in the previous step into the given equation. This substitution is a key step in solving the problem, as it allows us to combine the information we have about the arithmetic progression into a single equation. By substituting the expressions for a_5 and a_9, we eliminate these terms and obtain an equation that involves only a_1 and d. This equation represents a constraint on the relationship between the first term and the common difference. It tells us how these two parameters must be related in order for the given condition to be satisfied. This step demonstrates the power of algebraic manipulation in solving mathematical problems. By carefully substituting and simplifying expressions, we can transform a complex problem into a more manageable one. The resulting equation is a crucial piece of the puzzle, as it will allow us to find the sum of the first 13 terms of the arithmetic progression.

  3. Simplify the equation:

    2a_1 + 12d = 16

    a_1 + 6d = 8

    This step involves simplifying the equation obtained in the previous step. The goal is to make the equation more concise and easier to work with. By combining like terms and dividing both sides of the equation by a common factor, we arrive at a simpler equation that expresses the relationship between a_1 and d. This simplification is not just a cosmetic change; it makes the equation more transparent and easier to interpret. The simplified equation reveals a direct relationship between the first term and the common difference. It tells us that the sum of the first term and six times the common difference is equal to 8. This relationship is crucial for solving the problem, as it allows us to express one of the variables in terms of the other. This step highlights the importance of algebraic simplification in mathematical problem-solving. By simplifying equations, we can make them more manageable and gain a deeper understanding of the relationships between the variables.

  4. Recognize that a_7 = a_1 + 6d:

    This is the 7th term of the progression.

    This step is a crucial observation that connects the simplified equation (a_1 + 6d = 8) to the arithmetic progression. By recognizing that a_1 + 6d represents the 7th term of the progression (a_7), we can directly determine the value of a_7. This connection is not immediately obvious, but it is a key insight that simplifies the problem significantly. It demonstrates the importance of looking for patterns and connections within mathematical problems. By recognizing that a_1 + 6d is a specific term in the progression, we can bypass the need to solve for a_1 and d individually. This insight allows us to directly use the given information to find the sum of the first 13 terms. This step highlights the importance of mathematical intuition and the ability to see connections between different concepts. It demonstrates that problem-solving often involves more than just applying formulas; it requires a deep understanding of the underlying principles and the ability to make insightful observations.

  5. Therefore, a_7 = 8:

    This follows directly from the previous step and the simplified equation.

    This step is a direct consequence of the previous step. Since we recognized that a_1 + 6d represents the 7th term of the progression (a_7) and we know that a_1 + 6d = 8, we can conclude that a_7 = 8. This is a significant step in solving the problem, as we have now determined the value of a specific term in the progression. This value will be crucial for calculating the sum of the first 13 terms. This step demonstrates the power of logical deduction in mathematical problem-solving. By carefully combining information and applying logical reasoning, we can arrive at important conclusions. The determination of a_7 is a key milestone in the solution process, as it provides us with a crucial piece of information that we can use to find the sum.

  6. Use the formula for the sum of an arithmetic progression, S_n = (n/2) * (a_1 + a_n):

    We want to find S_13, so n = 13.

    This step involves recalling and applying the formula for the sum of an arithmetic progression. We are specifically using the formula S_n = (n/2) * (a_1 + a_n), which is particularly useful when we know the first term (a_1) and the last term (a_n). In this case, we want to find the sum of the first 13 terms (S_13), so n = 13. This step highlights the importance of having a repertoire of formulas and knowing when to apply them. The formula for the sum of an arithmetic progression is a powerful tool for calculating the sum of a sequence without having to add up all the individual terms. By recognizing that this formula is applicable to the problem at hand, we can proceed to the next step, where we will substitute the known values and solve for the sum.

  7. We can rewrite S_13 as S_13 = (13/2) * (a_1 + a_13):

    This is a direct application of the sum formula.

    This step is a straightforward application of the sum formula, substituting n = 13 into the general formula S_n = (n/2) * (a_1 + a_n). This substitution allows us to express the sum of the first 13 terms (S_13) in terms of the first term (a_1) and the 13th term (a_13). This is a necessary step in order to calculate the sum, as we need to know the values of a_1 and a_13. This step demonstrates the importance of being able to apply formulas correctly and to substitute values accurately. By rewriting the sum formula in this specific form, we are setting the stage for the next step, where we will use the information we have about the arithmetic progression to find the values of a_1 and a_13.

  8. Recognize that a_1 + a_13 = 2a_7:

    This is a key property of arithmetic progressions. The terms equidistant from the middle term have a sum equal to twice the middle term.

    This step is a crucial insight that leverages a key property of arithmetic progressions. In an arithmetic progression, the terms equidistant from the middle term have a sum equal to twice the middle term. In this case, a_1 and a_13 are equidistant from the 7th term (a_7), which is the middle term of the first 13 terms. Therefore, a_1 + a_13 = 2a_7. This property is a powerful tool for simplifying calculations and solving problems involving arithmetic progressions. By recognizing this property, we can avoid the need to calculate a_1 and a_13 individually. This step highlights the importance of understanding the properties of mathematical objects and how they can be used to simplify problem-solving. The recognition of this property is a key step in the solution, as it allows us to directly relate the sum of the first and 13th terms to the 7th term, which we have already determined.

  9. Substitute a_7 = 8:

    a_1 + a_13 = 2 * 8 = 16

    This step involves substituting the value of a_7, which we determined earlier to be 8, into the equation a_1 + a_13 = 2a_7. This substitution allows us to directly calculate the sum of the first and 13th terms. By substituting a_7 = 8, we find that a_1 + a_13 = 16. This is a significant result, as it provides us with the value of the sum of the first and 13th terms, which is a crucial component of the formula for the sum of the first 13 terms. This step demonstrates the importance of using previously obtained results to solve further steps in a problem. By leveraging the value of a_7, we can efficiently calculate the sum of the first and 13th terms.

  10. Substitute this back into the sum formula:

    S_13 = (13/2) * 16

    This step involves substituting the value of a_1 + a_13, which we just calculated to be 16, back into the formula for the sum of the first 13 terms: S_13 = (13/2) * (a_1 + a_13). This substitution allows us to express the sum of the first 13 terms in terms of a known value. By substituting a_1 + a_13 = 16, we obtain the equation S_13 = (13/2) * 16. This is a crucial step, as it reduces the problem to a simple arithmetic calculation. This step demonstrates the power of substitution in mathematical problem-solving. By carefully substituting known values into formulas and equations, we can simplify complex problems and arrive at solutions efficiently.

  11. Calculate S_13:

    S_13 = 13 * 8 = 104

    This final step involves performing the arithmetic calculation to find the value of S_13. By multiplying 13 by 8, we obtain the result S_13 = 104. This is the solution to the problem: the sum of the first 13 terms of the arithmetic progression is 104. This step demonstrates the importance of careful calculation and attention to detail in mathematical problem-solving. By accurately performing the arithmetic operation, we arrive at the correct answer. This final result represents the culmination of all the previous steps, demonstrating the power of a systematic and logical approach to problem-solving. The solution, S_13 = 104, provides a concrete answer to the original question and completes the solution process.

H2: Conclusion: The Sum Unveiled

Therefore, the sum of the first 13 terms of the arithmetic progression is 104. This problem demonstrates the power of combining the fundamental concepts and formulas related to arithmetic progressions. By systematically applying the general formula for the nth term and the formula for the sum of an arithmetic progression, we were able to solve the problem efficiently. This detailed solution provides a clear understanding of the steps involved and reinforces the importance of a logical and methodical approach to problem-solving. The ability to solve problems like this is crucial for developing a strong foundation in algebra and related mathematical fields. The key takeaways from this problem include the importance of understanding the general formula for the nth term, the formula for the sum of an arithmetic progression, and the properties of arithmetic progressions. By mastering these concepts, you can confidently tackle a wide range of problems involving arithmetic sequences and series. This problem also highlights the value of algebraic manipulation and simplification. By carefully simplifying equations and expressions, we can make problems more manageable and easier to solve. The ability to manipulate algebraic expressions is a fundamental skill in mathematics and is essential for success in higher-level courses. Furthermore, this problem demonstrates the importance of looking for connections and patterns within mathematical problems. By recognizing the relationship between the terms of an arithmetic progression and the properties of the sum, we were able to simplify the solution process. This ability to see connections and patterns is a hallmark of a strong mathematical thinker. In conclusion, this problem serves as an excellent example of how theoretical concepts can be applied to solve practical problems. It reinforces the importance of understanding the underlying principles of arithmetic progressions and their applications. By mastering these concepts and developing strong problem-solving skills, you can unlock the beauty and power of mathematics.