Explicit Formula for the Sequence 1/2, 2/3, 3/4 and Beyond

Have you ever come across a sequence where each term is a simple fraction, but the numerator and denominator have a simple numerical relationship? Understanding such sequences can greatly enhance your problem-solving skills, especially in fields like mathematics, computer science, and data analysis. In this article, we will explore the explicit formula for the sequence 1/2, 2/3, 3/4, and more. By the end, you will be able to express any term in the sequence using a single formula.

Understanding the Sequence

The sequence in question is 1/2, 2/3, 3/4, and so on. Let's break it down into simpler components:

Numerator: The numerator of each fraction is a simple sequence of integers: 1, 2, 3, 4, ... Denominator: The denominator of each fraction is one more than the numerator.

This relationship can be visually described as:

Numerator  nDenominator  n   1

Deriving the Explicit Formula

To find the explicit formula for the sequence, let's start by expressing each term in a more generalized form:

Term  n / (n   1)

For example:

When n 1, the term is 1 / (1 1) 1/2 When n 2, the term is 2 / (2 1) 2/3 When n 3, the term is 3 / (3 1) 3/4

This formula works for any integer value of n. Now, let's break it down even further:

Term  n / (n   1)  (n   1 - 1) / (n   1)  (n   1) / (n   1) - 1 / (n   1)  1 - 1 / (n   1)

This final form, 1 - 1 / (n 1), is the explicit formula we were looking for. It provides a straightforward way to determine the value of any term in the sequence.

Applications and Examples

The explicit formula for the sequence 1/2, 2/3, 3/4, and so on, can be applied in various scenarios. For instance, in programming or mathematical modeling, you can use this formula to generate the sequence or to quickly find the value of a specific term.

Example: Calculate the 10th term of the sequence.n  10Term  1 - 1 / (n   1)  1 - 1 / (10   1)  1 - 1/11  10/11

This is a simple yet powerful formula that can be used in a wide range of applications.

Conclusion

Understanding the explicit formula for the sequence 1/2, 2/3, 3/4, and so on, opens up numerous possibilities in various fields. With this knowledge, you can easily find any term in the sequence using the formula Term 1 - 1 / (n 1). Whether you are working on a mathematical problem or developing an algorithm, this formula will be a valuable tool in your arsenal.

If you have any questions or need further clarification, feel free to reach out. Happy exploring!