Finding The 12th Term From The End Of Arithmetic Progressions

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Finding a specific term from the end of an arithmetic progression (AP) is a common problem in mathematics. This article provides a detailed explanation of how to find the 12th term from the end of three different arithmetic progressions. We will explore the underlying concepts, formulas, and step-by-step solutions for each case. Understanding arithmetic progressions and their properties is crucial for solving such problems efficiently.

Understanding Arithmetic Progressions

Before diving into the solutions, let's define what an arithmetic progression is. 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:

  • a, a + d, a + 2d, a + 3d, ...

Where:

  • 'a' is the first term.
  • 'd' is the common difference.

The nth term of an AP can be calculated using the formula:

  • an = a + (n - 1)d

Where:

  • an is the nth term.
  • 'n' is the position of the term in the sequence.

To find a term from the end of an AP, we can either reverse the AP and then find the required term from the beginning or use a modified formula that directly calculates the term from the end. Both methods will be discussed in the solutions below. Let's proceed with the solutions for each of the given arithmetic progressions, ensuring clarity and providing step-by-step guidance for easy understanding. This approach helps in grasping the core concepts and applying them effectively to similar problems.

(i) AP: 3, 5, 7, 9, ..., 201

To find the 12th term from the end of the arithmetic progression 3, 5, 7, 9, ..., 201, we first need to identify the first term, the common difference, and the last term. This will allow us to use the appropriate formula to find the required term. Arithmetic progressions are sequences where the difference between consecutive terms remains constant, and understanding their properties is crucial for solving these types of problems. Let's break down the steps to solve this problem systematically.

Identifying Key Parameters

In the given AP, 3, 5, 7, 9, ..., 201:

  • The first term (a) is 3.
  • The common difference (d) can be calculated by subtracting the first term from the second term, or any term from its preceding term. So, d = 5 - 3 = 2.
  • The last term (l) is 201.

Now, we need to find the total number of terms in the AP to determine the 12th term from the end accurately. We can use the formula for the nth term of an AP to find the number of terms. The nth term formula is an = a + (n - 1)d, where an represents the nth term, a is the first term, n is the number of terms, and d is the common difference. By substituting the known values, we can find the value of n.

Finding the Total Number of Terms

Using the formula for the nth term:

  • an = a + (n - 1)d
  • 201 = 3 + (n - 1)2
  • 201 = 3 + 2n - 2
  • 201 = 1 + 2n
  • 200 = 2n
  • n = 100

Thus, there are 100 terms in the given arithmetic progression. Knowing the total number of terms is essential because it allows us to determine the position of the 12th term from the end when counted from the beginning. This conversion is a key step in solving this type of problem efficiently.

Determining the 12th Term from the End

To find the 12th term from the end, we can use the concept that the 12th term from the end is equivalent to the (100 - 12 + 1)th term from the beginning. This can be derived because if we count 12 terms from the end, we are left with 88 terms before them, plus the term itself, making it the 89th term from the start. So, we need to find the 89th term from the beginning.

  • Term from the beginning = n - (term from the end) + 1
  • Term from the beginning = 100 - 12 + 1 = 89

Now, we can use the nth term formula again to find the 89th term:

  • a89 = a + (89 - 1)d
  • a89 = 3 + (88)2
  • a89 = 3 + 176
  • a89 = 179

Therefore, the 12th term from the end of the AP 3, 5, 7, 9, ..., 201 is 179. This methodical approach, involving identifying key parameters, calculating the total number of terms, and then determining the required term, ensures accuracy and clarity in problem-solving. The final answer clearly demonstrates our understanding and application of arithmetic progression principles.

(ii) AP: 3, 8, 13, ..., 253 [NCERT]

To determine the 12th term from the end of the arithmetic progression 3, 8, 13, ..., 253, we will follow a similar approach to the previous problem. This involves identifying the key parameters of the AP, such as the first term, the common difference, and the last term. Understanding how these components fit together within the framework of an arithmetic progression is crucial for accurately finding the required term. Let’s systematically break down the problem and solve it step-by-step to ensure a clear and comprehensive understanding of the solution process.

Identifying Key Parameters

In the given AP, 3, 8, 13, ..., 253:

  • The first term (a) is 3.
  • The common difference (d) is the difference between consecutive terms. So, d = 8 - 3 = 5.
  • The last term (l) is 253.

Before we can find the 12th term from the end, we need to determine the total number of terms in the arithmetic progression. This is important because it helps us relate the position of the term from the end to its position from the beginning. We will use the nth term formula to find the number of terms in the AP, which is a crucial step in solving this problem.

Finding the Total Number of Terms

Using the formula for the nth term:

  • an = a + (n - 1)d
  • 253 = 3 + (n - 1)5
  • 253 = 3 + 5n - 5
  • 253 = 5n - 2
  • 255 = 5n
  • n = 51

So, there are 51 terms in this arithmetic progression. Knowing the total number of terms is a significant step as it allows us to calculate the position of the 12th term from the end when counted from the beginning. This step is vital for accurately determining the term we are looking for.

Determining the 12th Term from the End

To find the 12th term from the end, we determine its equivalent position from the beginning. The 12th term from the end corresponds to the (51 - 12 + 1)th term from the beginning. This conversion is necessary to use the standard nth term formula effectively. Let’s calculate the term number from the beginning:

  • Term from the beginning = n - (term from the end) + 1
  • Term from the beginning = 51 - 12 + 1 = 40

Therefore, we need to find the 40th term from the beginning of the AP. Now, using the nth term formula again:

  • a40 = a + (40 - 1)d
  • a40 = 3 + (39)5
  • a40 = 3 + 195
  • a40 = 198

Thus, the 12th term from the end of the AP 3, 8, 13, ..., 253 is 198. This solution demonstrates a step-by-step approach, from identifying parameters to finding the total terms and calculating the required term, ensuring clarity and accuracy in problem-solving. The final answer highlights the correct application of arithmetic progression principles.

(iii) AP: 1, 4, 7, 10, ..., 88

To find the 12th term from the end of the arithmetic progression 1, 4, 7, 10, ..., 88, we will follow a consistent method similar to the previous examples. This involves identifying the first term, the common difference, and the last term of the AP. These parameters are essential for using the nth term formula effectively. Understanding the relationship between these parameters within an arithmetic progression is key to solving the problem accurately. Let's break down the solution process into manageable steps for clarity and ease of understanding.

Identifying Key Parameters

In the given AP, 1, 4, 7, 10, ..., 88:

  • The first term (a) is 1.
  • The common difference (d) is the difference between consecutive terms. So, d = 4 - 1 = 3.
  • The last term (l) is 88.

Before finding the 12th term from the end, it's necessary to calculate the total number of terms in the AP. This helps us in relating the term's position from the end to its position from the beginning. We'll use the nth term formula to find the number of terms, which is a crucial step in solving this problem systematically.

Finding the Total Number of Terms

Using the formula for the nth term:

  • an = a + (n - 1)d
  • 88 = 1 + (n - 1)3
  • 88 = 1 + 3n - 3
  • 88 = 3n - 2
  • 90 = 3n
  • n = 30

Hence, there are 30 terms in the given arithmetic progression. Knowing the total number of terms is crucial for accurately determining the position of the 12th term from the end when counted from the beginning. This conversion is essential for solving the problem efficiently.

Determining the 12th Term from the End

To find the 12th term from the end, we first determine its equivalent position from the beginning. The 12th term from the end corresponds to the (30 - 12 + 1)th term from the beginning. This conversion allows us to use the standard nth term formula to find the required term. Let's calculate the term number from the beginning:

  • Term from the beginning = n - (term from the end) + 1
  • Term from the beginning = 30 - 12 + 1 = 19

So, we need to find the 19th term from the beginning of the AP. Now, using the nth term formula again:

  • a19 = a + (19 - 1)d
  • a19 = 1 + (18)3
  • a19 = 1 + 54
  • a19 = 55

Thus, the 12th term from the end of the AP 1, 4, 7, 10, ..., 88 is 55. This solution showcases a structured approach, identifying key parameters, calculating total terms, and determining the specific term, ensuring a clear and accurate solution. The final answer confirms the correct application of arithmetic progression principles.

Conclusion

In this article, we have explored how to find a specific term from the end of an arithmetic progression. By understanding the basic concepts of APs, identifying key parameters, and using the appropriate formulas, we can solve these problems efficiently. The step-by-step solutions provided for each case demonstrate a systematic approach to problem-solving, emphasizing the importance of clarity and accuracy. Whether it’s finding the total number of terms or determining the position of a term from the end, each step is crucial in arriving at the correct answer. By mastering these techniques, you can confidently tackle a variety of arithmetic progression problems.