Arithmetic Progression Nth Term Formula And Solving Cost Equations
In the realm of mathematics, arithmetic progressions hold a significant place, providing a structured sequence of numbers that follow a predictable pattern. At the heart of an arithmetic progression lies the concept of a common difference, the constant value added to each term to obtain the next. This consistent increment allows us to define and analyze these sequences with precision. One of the fundamental tasks when dealing with arithmetic progressions is determining the value of a specific term within the sequence, often referred to as the nth term. This is where the power of mathematical formulas comes into play, providing a direct and efficient way to calculate the nth term without having to manually compute all the preceding terms. The formula for the nth term of an arithmetic progression is a cornerstone concept, enabling us to solve various problems related to these sequences. It empowers us to predict future terms, analyze patterns, and make informed decisions based on the underlying arithmetic progression.
The formula itself is elegantly simple yet remarkably powerful: an = a + (n - 1)d. Let's dissect this formula to understand its components and how they interact. The symbol an represents the nth term, the very value we are trying to find. The letter 'a' denotes the first term of the arithmetic progression, the starting point of our sequence. The variable 'n' signifies the term number, indicating which position in the sequence we are interested in. Finally, 'd' represents the common difference, the constant value added to each term. The formula essentially states that the nth term is equal to the first term plus the common difference multiplied by one less than the term number. This seemingly simple equation encapsulates the essence of arithmetic progressions, allowing us to navigate through the sequence with ease.
To illustrate the application of this formula, consider an arithmetic progression with a first term of 5 and a common difference of 3. If we want to find the 10th term, we simply plug the values into the formula: a10 = 5 + (10 - 1) * 3. This simplifies to a10 = 5 + 9 * 3, which further reduces to a10 = 5 + 27. Therefore, the 10th term of this arithmetic progression is 32. This example demonstrates the directness and efficiency of the formula, allowing us to calculate the nth term without having to list out all the preceding terms. The formula for the nth term is not just a mathematical tool; it's a gateway to understanding and manipulating arithmetic progressions, empowering us to solve a wide range of problems and explore the patterns within these sequences.
Algebraic equations are the language of problem-solving, providing a framework for representing relationships between known and unknown quantities. In this scenario, we encounter a practical problem involving the cost of hats and balls, where we need to leverage algebraic techniques to determine the unknown cost of a single ball. The given information provides us with a crucial equation: 3 hats + 2 balls = Rs. 960. This equation encapsulates the total cost of the purchase, but it also introduces an unknown – the individual cost of a ball. To unravel this mystery, we need to utilize the power of algebraic manipulation and substitution.
The additional piece of information, the cost of a bat (Rs. 300), might seem unrelated at first glance. However, it serves as a vital clue that allows us to simplify the problem. The mention of the bat's cost subtly indicates that there might be a typo in the original problem statement. It is highly likely that the word "bat" was mistakenly written instead of "hat." If we assume this correction, the problem transforms into a more solvable scenario. With this assumption, we can rephrase the given information as: 3 hats + 2 balls = Rs. 960, and the cost of one hat is Rs. 300. This seemingly small correction opens up a clear pathway to solving for the cost of a ball.
Now, we can use substitution to find the cost of the balls. Since we know the cost of one hat is Rs. 300, the cost of 3 hats would be 3 * Rs. 300 = Rs. 900. Substituting this value back into the original equation, we get: Rs. 900 + 2 balls = Rs. 960. To isolate the cost of the balls, we subtract Rs. 900 from both sides of the equation, resulting in: 2 balls = Rs. 60. Finally, to find the cost of one ball, we divide both sides of the equation by 2: 1 ball = Rs. 30. Therefore, the cost of one ball is Rs. 30. This problem exemplifies the power of algebraic equations in translating real-world scenarios into mathematical expressions and using systematic techniques to solve for unknown quantities. By carefully analyzing the given information, identifying potential errors, and applying algebraic principles, we successfully determined the cost of a ball.
In conclusion, both the formula for the nth term of an arithmetic progression and the methods for solving algebraic equations are fundamental tools in mathematics. The arithmetic progression formula provides a direct way to calculate any term in a sequence, while algebraic equations allow us to model and solve real-world problems involving unknown quantities. Mastering these concepts is essential for building a strong foundation in mathematics and its applications.
