The 3rd Term Of An AP Is 4, The 15th Term Is 8, And The Last Term Is 18. Determine If The Following Statements Are Correct: 1. The Common Difference Is \( \frac{1}{3} \). 2. The Number Of Terms Is 45.

This article delves into solving a classic arithmetic progression (AP) problem. We will explore how to determine the common difference and the number of terms in an AP when given specific terms. Arithmetic progressions are fundamental in mathematics, appearing in various applications, from simple sequences to complex mathematical models. Understanding how to solve problems related to APs is crucial for students and anyone interested in mathematics. In this problem, we are given that the 3rd, 15th, and the last terms of an arithmetic progression are 4, 8, and 18, respectively. Our goal is to determine the common difference and the number of terms in this AP. We will analyze the given information step by step, applying the formulas and properties of arithmetic progressions to arrive at the solution. This article aims to provide a clear and detailed explanation of the solution process, making it easy for readers to understand and apply the concepts to similar problems. By breaking down the problem into manageable parts, we will demonstrate how to use the information effectively and arrive at the correct conclusions. The concepts covered here are essential for anyone studying sequences and series in mathematics, and the problem-solving approach can be applied to a wide range of similar problems.

Understanding Arithmetic Progressions

To effectively tackle this problem, it is essential to first understand the basics of arithmetic progressions. An arithmetic progression 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, often denoted by 'd'. The general form of an AP is given by: a, a + d, a + 2d, a + 3d, and so on, where 'a' is the first term. The nth term of an AP, denoted by aₙ, can be calculated using the formula: aₙ = a + (n - 1)d. This formula is crucial for solving problems related to APs, as it allows us to find any term in the sequence if we know the first term, the common difference, and the position of the term. The sum of the first n terms of an AP can be found using the formula: Sₙ = n/2 [2a + (n - 1)d], or Sₙ = n/2 (a + l), where 'l' is the last term. Understanding these formulas and properties is essential for solving various problems involving arithmetic progressions. In this particular problem, we are given specific terms of the AP, and we need to find the common difference and the number of terms. By applying the formula for the nth term, we can set up equations and solve for the unknowns. The key is to carefully analyze the given information and use the formulas strategically to arrive at the solution. This article will guide you through the step-by-step process of applying these concepts to solve the given problem effectively. Remember, a strong foundation in the basic principles of APs is crucial for tackling more complex problems in sequences and series.

Problem Statement

The problem states that the 3rd term of an arithmetic progression is 4, the 15th term is 8, and the last term is 18. We are presented with two statements:

1. The common difference is ${ \frac{1}{3} }$
2. The number of terms is 45

Our task is to determine which of these statements, if any, are correct. This requires us to use the properties of arithmetic progressions and the given information to calculate the common difference and the number of terms. We will start by using the formula for the nth term of an AP to set up equations based on the given information. Then, we will solve these equations to find the common difference. Once we have the common difference, we can use the information about the last term to determine the number of terms in the AP. This involves a careful application of the formulas and algebraic manipulation. It is essential to keep track of the information and use it systematically to avoid errors. The problem tests our understanding of arithmetic progressions and our ability to apply the formulas correctly. By breaking the problem down into smaller steps, we can make the solution process more manageable and increase our chances of arriving at the correct answer. The challenge lies in correctly setting up the equations and solving them accurately. This article will guide you through each step, providing clear explanations and justifications to help you understand the process thoroughly. Remember, a systematic approach is key to solving problems like this efficiently.

Solving for the Common Difference

To find the common difference, we will use the information about the 3rd and 15th terms of the AP. Let 'a' be the first term and 'd' be the common difference. The formula for the nth term of an AP is aₙ = a + (n - 1)d. According to the problem, the 3rd term (a₃) is 4 and the 15th term (a₁₅) is 8. Using the formula, we can write two equations:

1. a₃ = a + 2d = 4
2. a₁₅ = a + 14d = 8

Now, we have a system of two linear equations with two variables (a and d). We can solve this system to find the values of 'a' and 'd'. One common method for solving such systems is the elimination method. We can subtract the first equation from the second equation to eliminate 'a': (a + 14d) - (a + 2d) = 8 - 4. This simplifies to 12d = 4. Dividing both sides by 12, we get d = ${ \frac{4}{12} }$ = ${ \frac{1}{3} }$. Therefore, the common difference is ${ \frac{1}{3} }$. This confirms the first statement that the common difference is ${ \frac{1}{3} }$. The process of solving for the common difference involved setting up equations based on the formula for the nth term and then using algebraic manipulation to solve the system of equations. This is a common technique in solving problems related to arithmetic progressions. By carefully applying the formula and using algebraic techniques, we were able to find the value of the common difference. This step is crucial for solving the rest of the problem, as we will use the common difference to find the number of terms in the AP. Understanding how to solve for the common difference is essential for mastering arithmetic progression problems.

Determining the First Term

Now that we have found the common difference, d = ${ \frac{1}{3} }$, we can determine the first term 'a'. We can use either of the equations we set up earlier. Let's use the equation for the 3rd term: a + 2d = 4. Substituting d = ${ \frac{1}{3} }$ into this equation, we get: a + 2(${ \frac{1}{3} }$) = 4. This simplifies to a + ${ \frac{2}{3} }$ = 4. To solve for 'a', we subtract ${ \frac{2}{3} }$ from both sides: a = 4 - ${ \frac{2}{3} }$. Converting 4 to a fraction with a denominator of 3, we have a = ${ \frac{12}{3} }$ - ${ \frac{2}{3} }$, which gives us a = ${ \frac{10}{3} }$. Thus, the first term of the arithmetic progression is ${ \frac{10}{3} }$. Finding the first term is an important step in solving AP problems, as it allows us to fully define the sequence. With the first term and the common difference known, we can find any term in the AP using the formula aₙ = a + (n - 1)d. In this case, knowing the first term and the common difference will help us determine the number of terms in the AP, which is the next step in solving the problem. The process of finding the first term involved substituting the known common difference into one of the equations derived from the given information. This demonstrates the interconnectedness of the elements in an arithmetic progression. By systematically solving for the unknowns, we can build a complete picture of the sequence. This step-by-step approach is crucial for solving more complex problems in mathematics.

Calculating the Number of Terms

To calculate the number of terms in the AP, we use the information about the last term. We are given that the last term is 18. Let's denote the number of terms as 'n'. The nth term of the AP is given by aₙ = a + (n - 1)d. Since the last term is 18, we have: 18 = a + (n - 1)d. We already know that a = ${ \frac{10}{3} }$ and d = ${ \frac{1}{3} }$. Substituting these values into the equation, we get: 18 = ${ \frac{10}{3} }$ + (n - 1)(${ \frac{1}{3} }$). To solve for 'n', we first subtract ${ \frac{10}{3} }$ from both sides: 18 - ${ \frac{10}{3} }$ = (n - 1)(${ \frac{1}{3} }$). Converting 18 to a fraction with a denominator of 3, we have ${ \frac{54}{3} }$ - ${ \frac{10}{3} }$ = (n - 1)(${ \frac{1}{3} }$). This simplifies to ${ \frac{44}{3} }$ = (n - 1)(${ \frac{1}{3} }$). Next, we multiply both sides by 3 to get rid of the fraction: 44 = n - 1. Adding 1 to both sides, we find n = 45. Therefore, the number of terms in the arithmetic progression is 45. This confirms the second statement that the number of terms is 45. Calculating the number of terms involved using the formula for the nth term and substituting the known values of the first term, common difference, and the last term. This is a common method for finding the number of terms in an AP when the last term is known. By carefully applying the formula and using algebraic manipulation, we were able to find the value of 'n'. This completes our analysis of the arithmetic progression, as we have now found both the common difference and the number of terms.

Conclusion

In conclusion, we have analyzed the given arithmetic progression problem and determined the common difference and the number of terms. We found that the common difference is ${ \frac{1}{3} }$ and the number of terms is 45. Therefore, both statements provided in the problem are correct. This problem demonstrates the application of the formulas and properties of arithmetic progressions. By using the formula for the nth term, aₙ = a + (n - 1)d, we were able to set up equations based on the given information and solve for the unknowns. The process involved solving a system of linear equations to find the common difference and then using the information about the last term to calculate the number of terms. This step-by-step approach is crucial for solving problems related to arithmetic progressions. The key is to carefully analyze the given information, identify the relevant formulas, and apply them systematically. This article has provided a detailed explanation of the solution process, making it easy for readers to understand and apply the concepts to similar problems. Understanding arithmetic progressions is essential for anyone studying sequences and series in mathematics. The problem-solving techniques demonstrated here can be applied to a wide range of similar problems. By mastering these techniques, you can confidently tackle more complex problems in mathematics. Remember, a strong foundation in the basic principles and a systematic approach are key to success in problem-solving. The ability to break down a problem into smaller, manageable steps is crucial for arriving at the correct solution. We hope this article has been helpful in enhancing your understanding of arithmetic progressions and problem-solving.

Final Answer

(C) Both 1 and 2