Solving A Recurrence Of The Form U N A N = ∑ K = 0 N − 1 C K A K U_n A_n = \sum_{k=0}^{n-1} C_k A_k U N A N = ∑ K = 0 N − 1 C K A K

May 22, 2025 by ADMIN 142 views

$Solving a Recurrence of the Form $u_n a_n = \sum_{k=0}^{n-1} c_k a_k$$

Introduction

When dealing with recurrence relations, it's not uncommon to encounter equations of the form $u_n a_n = \sum_{k=0}^{n-1} c_k a_k$ . Here, both $u_n$ and $a_n$ are sequences depending on $n$ , while the $c_k$ 's are constants that are used to weight the terms in the sum. In this article, we'll explore how to solve such recurrence relations for $a_n$ , the sequence we're interested in.

Understanding the Recurrence Relation

Before diving into the solution, let's take a closer look at the recurrence relation. We have:

u_n a_n = \sum_{k=0}^{n-1} c_k a_k

Here, $u_n$ is a sequence that depends on $n$ , and $a_n$ is the sequence we're trying to solve for. The $c_k$ 's are constants that are used to weight the terms in the sum. The sum is taken over all $k$ from $0$ to $n-1$ .

Solving the Recurrence Relation

To solve the recurrence relation, we can start by rearranging the equation to isolate $a_n$ :

a_n = \frac{1}{u_n} \sum_{k=0}^{n-1} c_k a_k

This equation still looks complicated, but we can simplify it further by using a technique called "telescoping". The idea behind telescoping is to cancel out terms in the sum, leaving only the first and last terms.

Telescoping the Sum

To telescope the sum, we can rewrite it as:

\sum_{k=0}^{n-1} c_k a_k = \sum_{k=0}^{n-1} c_k a_k + c_n a_n - c_n a_n

This looks like a mess, but it's actually a clever trick. By adding and subtracting $c_n a_n$ , we've created a term that cancels out with the $c_n a_n$ term in the original sum.

Simplifying the Sum

Now that we've telescoped the sum, we can simplify it further:

\sum_{k=0}^{n-1} c_k a_k = \sum_{k=0}^{n-2} c_k a_k + c_{n-1} a_{n-1}

This is a much simpler sum, and we can see that the terms are canceling out nicely.

Solving for $a_n$

Now that we've simplified the sum, we can solve for $a_n$ :

a_n = \frac{1}{u_n} \sum_{k=0}^{n-2} c_k a_k + \frac{c_{n-1} a_{n-1}}{u_n}

This is the solution to the recurrence relation, and it's a much simpler equation than the original one.

Example

Let's consider an example to illustrate how to use this solution. Suppose we have the recurrence relation:

2a_n = a_{n-1} + 3a_{n-2}

We can rewrite this as:

a_n = \frac{1}{2} a_{n-1} + \frac{3}{2} a_{n-2}

Using the solution we derived earlier, we can write:

a_n = \frac{1}{2} \sum_{k=0}^{n-2} c_k a_k + \frac{c_{n-1} a_{n-1}}{2}

where $c_k = 1$ for all $k$ .

Conclusion

In this article, we've explored how to solve recurrence relations of the form $u_n a_n = \sum_{k=0}^{n-1} c_k a_k$ . We've used the technique of telescoping to simplify the sum, and we've derived a solution for $a_n$ . We've also illustrated how to use this solution with an example. By following these steps, you should be able to solve similar recurrence relations and gain a deeper understanding of the underlying mathematics.

References

[1] Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to algorithms. MIT Press.

[2] MIT OpenCourseWare. (n.d.). Recurrence Relations. Retrieved from https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-introduction-to-algorithms-spring-2005/lecture-notes/lec6.pdf

[3] Wolfram MathWorld. (n.d.). Recurrence Relations. Retrieved from https://mathworld.wolfram.com/RecurrenceRelation.html

Q&A: Solving Recurrence Relations

In the previous article, we explored how to solve recurrence relations of the form $u_n a_n = \sum_{k=0}^{n-1} c_k a_k$ . In this article, we'll answer some common questions about solving recurrence relations and provide additional examples to illustrate the concepts.

Q: What is a recurrence relation?

A: A recurrence relation is an equation that defines a sequence recursively. It's a way of describing a sequence in terms of its previous values.

Q: What is the difference between a recurrence relation and a recursive function?

A: A recurrence relation is an equation that defines a sequence recursively, while a recursive function is a function that calls itself to compute its value. While related, these two concepts are not the same.

Q: How do I know if a recurrence relation can be solved using the method we described?

A: To determine if a recurrence relation can be solved using the method we described, you need to check if the relation can be written in the form $u_n a_n = \sum_{k=0}^{n-1} c_k a_k$ . If it can, then you can use the method to solve the relation.

Q: What if the recurrence relation has multiple terms on the right-hand side?

A: If the recurrence relation has multiple terms on the right-hand side, you can still use the method we described. However, you'll need to use a more general approach to solve the relation.

Q: Can I use the method we described to solve recurrence relations with negative indices?

A: No, the method we described is only applicable to recurrence relations with non-negative indices.

Q: How do I choose the values of $c_k$ ?

A: The values of $c_k$ are typically chosen to make the recurrence relation as simple as possible. In some cases, the values of $c_k$ may be determined by the problem itself.

Q: Can I use the method we described to solve recurrence relations with complex coefficients?

A: Yes, the method we described can be used to solve recurrence relations with complex coefficients.

Q: How do I know if the solution to the recurrence relation is unique?

A: The solution to a recurrence relation is unique if the relation is linear and the initial conditions are specified.

Q: Can I use the method we described to solve recurrence relations with multiple variables?

A: No, the method we described is only applicable to recurrence relations with a single variable.

Example 1: Solving a Recurrence Relation with a Constant Coefficient

Suppose we have the recurrence relation:

a_n = 2a_{n-1} + 3