Finding A_n In An Arithmetic Progression When A_{n+1} Equals 4n + 5
Arithmetic progressions, also known as arithmetic sequences, are fundamental concepts in mathematics. Understanding these sequences involves recognizing patterns and relationships between consecutive terms. In this article, we will delve into a specific problem involving an arithmetic progression, where the (n+1)th term is given by a_{n+1} = 4n + 5, and our goal is to determine the expression for the nth term, a_n. This exploration will provide a deeper understanding of how arithmetic progressions work and how to manipulate their formulas to find specific terms.
Defining 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 called the common difference, often denoted by 'd'. The general form of an arithmetic progression is:
a, a + d, a + 2d, a + 3d, ...
where 'a' is the first term of the sequence. The nth term of an arithmetic progression can be expressed as:
a_n = a + (n - 1)d
This formula is crucial for solving problems related to arithmetic progressions as it allows us to find any term in the sequence if we know the first term and the common difference. Understanding this formula is the cornerstone to unraveling the complexities of arithmetic progressions.
Problem Statement: Given a_{n+1} = 4n + 5, Find a_n
The problem at hand presents us with a unique challenge. We are given the expression for the (n+1)th term of an arithmetic progression, a_{n+1} = 4n + 5, and we are tasked with finding the expression for the nth term, a_n. This requires us to manipulate the given expression and apply the principles of arithmetic progressions to arrive at the correct solution. The options provided are:
(A) 4n + 1 (B) 4n - 1 (C) 4n - 5 (D) 4n + 5
To solve this, we need to understand how the terms of an arithmetic progression relate to each other and how the common difference plays a role in determining the terms. The process involves careful substitution and algebraic manipulation to derive the expression for a_n.
Step 1: Expressing a_n in terms of a_{n+1}
To find a_n, we need to relate it to the given expression for a_{n+1}. We know that in an arithmetic progression, the difference between consecutive terms is constant. Therefore, we can express a_n in terms of a_{n+1} and the common difference, d. The relationship between a_n and a_{n+1} can be written as:
a_{n+1} = a_n + d
This equation is the key to our solution. It tells us that the (n+1)th term is equal to the nth term plus the common difference. We can rearrange this equation to solve for a_n:
a_n = a_{n+1} - d
Now, we need to find the common difference, d, to substitute it into this equation.
Step 2: Determining the Common Difference, d
The common difference, d, is the constant difference between consecutive terms in an arithmetic progression. To find d, we can consider two consecutive terms, a_{n+1} and a_n, and subtract them. However, since we only have the expression for a_{n+1}, we need to find another term to compare with. We can find a_n by replacing 'n' with 'n-1' in the expression for a_{n+1}:
a_{n+1} = 4n + 5
Replacing 'n' with 'n-1':
a_{(n-1)+1} = a_n = 4(n-1) + 5
Simplifying the expression for a_n:
a_n = 4n - 4 + 5
a_n = 4n + 1
Now that we have an expression for a_n, we can find the common difference, d, by subtracting a_n from a_{n+1}:
d = a_{n+1} - a_n
Substituting the expressions for a_{n+1} and a_n:
d = (4n + 5) - (4n + 1)
Simplifying the expression for d:
d = 4n + 5 - 4n - 1
d = 4
Thus, the common difference, d, is 4. This constant difference is what defines the arithmetic nature of the sequence and is crucial for finding any term within the progression.
Step 3: Substituting d into the Equation for a_n
Now that we have the common difference, d = 4, we can substitute it back into the equation we derived in Step 1:
a_n = a_{n+1} - d
We know that a_{n+1} = 4n + 5, so we substitute this into the equation:
a_n = (4n + 5) - 4
Simplifying the expression for a_n:
a_n = 4n + 5 - 4
a_n = 4n + 1
Therefore, the expression for the nth term, a_n, is 4n + 1. This result confirms that the nth term can be expressed as a linear function of n, which is characteristic of arithmetic progressions. This step-by-step approach ensures that we arrive at the correct expression for a_n by carefully considering the properties of arithmetic progressions and the given information.
After carefully analyzing the given information and applying the principles of arithmetic progressions, we have successfully determined the expression for the nth term, a_n. The steps involved expressing a_n in terms of a_{n+1}, finding the common difference, d, and substituting it back into the equation. Our calculations have led us to the following result:
a_n = 4n + 1
This corresponds to option (A) in the problem statement. Therefore, the correct answer is:
(A) 4n + 1
This exercise highlights the importance of understanding the fundamental properties of arithmetic progressions and how to manipulate their formulas to solve problems. The ability to express terms in relation to each other and to find the common difference is crucial in determining any term in the sequence. This problem serves as a valuable learning experience for anyone studying arithmetic progressions and their applications in mathematics.
Further Exploration of Arithmetic Progressions
Understanding arithmetic progressions goes beyond just finding individual terms. There are many other aspects to explore, such as the sum of the first n terms, applications in real-world scenarios, and more complex problems involving multiple arithmetic progressions. By delving deeper into these topics, one can gain a more comprehensive understanding of arithmetic progressions and their significance in mathematics and various other fields. The exploration of these sequences provides a foundation for understanding more advanced mathematical concepts and problem-solving techniques.
