In An Arithmetic Progression, The First Term Is 1 And The Sum Of The Nth Term With The Number Of Terms Is 2. Calculate The Common Difference Of This Progression.

Jun 19, 2025 by ADMIN 162 views

Calculating the Common Difference in an Arithmetic Progression: A Step-by-Step Guide

Understanding arithmetic progressions is fundamental in mathematics, and this article delves into a specific problem involving an arithmetic progression (AP). We'll break down the problem step-by-step, making it easy to grasp even if you're new to the concept. Our goal is to calculate the common difference of an AP given certain conditions. Let's get started!

Understanding Arithmetic Progressions

Before diving into the problem, let's quickly recap what an arithmetic progression is. An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by 'r'. The first term of the sequence is usually denoted by 'a₁'. So, a typical AP looks like this: a₁, a₁ + r, a₁ + 2r, a₁ + 3r, and so on.

Each term in the sequence can be represented by the formula: aₙ = a₁ + (n - 1)r, where aₙ is the nth term, a₁ is the first term, n is the term number, and r is the common difference. This formula is crucial for solving problems related to arithmetic progressions.

The sum of the first n terms of an arithmetic progression, denoted by Sₙ, is given by the formula: Sₙ = (n/2) * [2a₁ + (n - 1)r]. This formula provides a quick way to find the sum without having to add each term individually. Understanding these basics is key to tackling more complex problems.

Moreover, the concept of arithmetic progression extends beyond simple numerical sequences. It appears in various mathematical contexts and real-world applications, such as linear functions, financial calculations (like simple interest), and even in patterns observed in nature. Therefore, mastering arithmetic progressions is not just an academic exercise but a valuable skill with practical implications.

Problem Statement

The problem presented is a classic example of arithmetic progression questions. It challenges us to find the common difference (r) given specific information about the progression. The problem states: In an arithmetic progression, the first term (a₁) is 1, and the sum of the nth term (aₙ) and the number of terms (n) is 2. Our task is to calculate the common difference (r) of this progression.

This type of problem often requires us to use the formulas and properties of arithmetic progressions to set up equations and solve for the unknown variable, which in this case is 'r'. The challenge lies in translating the given information into mathematical expressions and then manipulating those expressions to isolate 'r'. A systematic approach, combined with a solid understanding of the formulas, is essential for solving such problems effectively. The key is to identify the relationships between the given quantities and the unknown, and then choose the appropriate formulas to bridge the gap. So, let's proceed by expressing the given information in mathematical terms and see how we can find the common difference.

To solve this, we'll need to translate the words into mathematical expressions. We know a₁ = 1, and aₙ + n = 2. The goal is to find 'r'. This problem highlights the importance of understanding the relationships between different elements of an arithmetic progression and using the given information strategically to find what's asked.

Setting up the Equations

Now, let's translate the given information into mathematical equations. We are given that the first term, a₁, is 1. We can write this as:

a₁ = 1

We are also given that the sum of the nth term (aₙ) and the number of terms (n) is 2. This can be written as:

aₙ + n = 2

We also know the general formula for the nth term of an arithmetic progression:

aₙ = a₁ + (n - 1)r

Our goal is to find 'r', the common difference. We now have three equations, and we can use these to solve for 'r'. The key is to substitute and manipulate these equations to eliminate variables and isolate 'r'. This involves algebraic techniques such as substitution and simplification, which are fundamental to solving mathematical problems. The ability to set up equations correctly from the problem statement is a crucial skill in problem-solving. It allows us to translate a conceptual understanding of the problem into a concrete mathematical framework that can be solved using standard techniques.

The next step is to use these equations to find a way to express 'r' in terms of the known quantities. We can start by substituting the value of a₁ from equation (1) into equation (3), and then use equation (2) to further simplify and solve for 'r'.

Solving for the Common Difference (r)

Let's proceed with solving for the common difference 'r'. We have the equations:

a₁ = 1
aₙ + n = 2
aₙ = a₁ + (n - 1)r

First, substitute a₁ = 1 into equation (3):

aₙ = 1 + (n - 1)r

Now, from equation (2), we have aₙ = 2 - n. Substitute this into the above equation:

2 - n = 1 + (n - 1)r

Next, simplify the equation and solve for 'r'. Subtract 1 from both sides:

1 - n = (n - 1)r

Now, divide both sides by (n - 1):

r = (1 - n) / (n - 1)

Notice that (1 - n) is the negative of (n - 1), so we can rewrite this as:

r = -(n - 1) / (n - 1)

As long as n ≠ 1, we can cancel out the (n - 1) terms:

r = -1

Thus, the common difference 'r' is -1. This solution demonstrates the power of algebraic manipulation and substitution in solving mathematical problems. By carefully applying the given information and the formulas of arithmetic progressions, we were able to isolate the unknown variable and find its value. The condition n ≠ 1 is important because division by zero is undefined. However, in the context of an arithmetic progression, if n = 1, the problem wouldn't make sense as we wouldn't have a progression.

Analyzing the Result

We have found that the common difference, r, is -1. Now, let's analyze this result in the context of the problem. This means that for each term in the arithmetic progression, we subtract 1 from the previous term. Given that the first term (a₁) is 1, the progression would look like this: 1, 0, -1, -2, and so on.

To verify our result, let's consider the second piece of information given in the problem: aₙ + n = 2. If r = -1, then aₙ = a₁ + (n - 1)r = 1 + (n - 1)(-1) = 1 - n + 1 = 2 - n. Therefore, aₙ + n = (2 - n) + n = 2, which matches the given condition.

This analysis confirms that our calculated value for the common difference is correct. It also illustrates the importance of checking the solution against the original problem statement to ensure consistency and accuracy. Understanding the implications of the solution in the context of the problem is a crucial step in mathematical problem-solving.

Furthermore, this result provides insight into the nature of arithmetic progressions. A negative common difference indicates that the terms of the progression are decreasing. This understanding helps in visualizing the progression and predicting the behavior of its terms.

Conclusion

In conclusion, by carefully applying the formulas and properties of arithmetic progressions, we successfully calculated the common difference (r) for the given problem. We found that r = -1. This exercise demonstrates the importance of understanding the fundamental concepts of arithmetic progressions and the ability to translate word problems into mathematical equations. The process involved setting up equations based on the given information, using substitution and algebraic manipulation to solve for the unknown variable, and finally, analyzing the result in the context of the problem.

This step-by-step approach is valuable for solving a wide range of mathematical problems. It emphasizes the importance of a systematic methodology, which includes understanding the problem statement, identifying the relevant formulas, setting up equations, solving for the unknowns, and verifying the solution. Mastering these skills is essential for success in mathematics and related fields.

The specific problem we solved highlights the relationship between the terms, common difference, and the number of terms in an arithmetic progression. It also reinforces the concept that a negative common difference leads to a decreasing sequence. By working through this example, we have gained a deeper understanding of arithmetic progressions and the techniques used to solve problems involving them.

Final Answer: The final answer is $\boxed{-1}$