Notes/Recurrence relations.md at a9676272f2a299bd2a0bf4a256462db3a605375b

2024-12-07 21:07:38 +01:00

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A recurrence relation is an equation that defines a sequence based on its earlier terms, along with initial values.

The recurrence relation a_n = a_{n-1} + 4 with initial condition a_1 = 3 defines the sequence: 3, 7, 11, 15, \ldots.

Techniques for Finding Explicit Formulas

Backtracking involves repeatedly substituting the recurrence relation into itself until a pattern emerges.
- For the recurrence relation a_n = a_{n-1} + 4, we repeatedly substitute:
  - a_n = a_{n-1} + 4
  - a_n = (a_{n-2} + 4) + 4 = a_{n-2} + 2 \cdot 4
  - a_n = ((a_{n-3} + 4) + 4) + 4 = a_{n-3} + 3 \cdot 4
  - \ldots
  - a_n = a_{n-(n-1)} + (n-1) \cdot 4 = a_1 + (n-1) \cdot 4 = 3 + (n-1) \cdot 4
- So, the explicit formula for the sequence is:
```
a_n = 3 + (n-1) \cdot 4
```
Characteristic Equation applies to linear homogeneous recurrence relations.
- A LHR relation of degree k is of the form:
```
s_n = a_1 s_{n-1} + a_2 s_{n-2} + \ldots + a_k s_{n-k},
```
  where a_i \in \mathbb{R} are constants. ¹
- The characteristic equation is:
```
x^k - a_1 x^{k-1} - a_2 x^{k-2} - \ldots - a_k = 0
```
- The roots of the characteristic equation determine the explicit formula for the sequence. The sources focus on degree-2 relations, but the method generalizes to any degree.

Solving Linear Homogeneous Recurrence Relations of Degree 2

For s_n = a s_{n-1} + b s_{n-2}, the characteristic equation is x^2 - ax - b = 0. Let r_1 and r_2 be the roots:

In case of distinct roots (r_1 \neq r_2):
- The general solution is:
```
s_n = c_1 r_1^n + c_2 r_2^n,
```
  where c_1 and c_2 are constants determined by initial conditions.
In case of repeated roots (r_1 = r_2 = r):
- The general solution is:
```
s_n = r^n (c_1 + c_2 n),
```
  where c_1 and c_2 are constants determined by initial conditions. ²

Example - Fibonacci Sequence

The Fibonacci sequence is defined as:


f_n = 
\begin{cases}
0, & \text{if } n = 0, \\
1, & \text{if } n = 1, \\
f_{n-1} + f_{n-2}, & \text{if } n \geq 2.
\end{cases}

The characteristic equation is:
```
x^2 - x - 1 = 0
```

The roots are:


r_1 = \frac{1 + \sqrt{5}}{2}, \quad r_2 = \frac{1 - \sqrt{5}}{2}.

Since r_1 \neq r_2, the explicit formula is:
```
f_n = c_1 r_1^n + c_2 r_2^n.
```
Using the initial conditions:
- f_0 = 0 = c_1 + c_2
- f_1 = 1 = c_1 \left(\frac{1 + \sqrt{5}}{2}\right) + c_2 \left(\frac{1 - \sqrt{5}}{2}\right)

Solving this system, we get:


c_1 = \frac{1}{\sqrt{5}}, \quad c_2 = -\frac{1}{\sqrt{5}}.

Therefore, the explicit formula for the Fibonacci sequence is:


f_n = \frac{1}{\sqrt{5}} \left(\frac{1 + \sqrt{5}}{2}\right)^n - \frac{1}{\sqrt{5}} \left(\frac{1 - \sqrt{5}}{2}\right)^n.

Verifying Explicit Formulas

The correctness of an explicit formula for a recurrence relation can be proven using strong mathematical induction. For example, the explicit Fibonacci formula is verified by Induction.

Footnotes

Linear homogeneous recurrence relations are equations where each term is a combination of earlier terms, with no added constants. ↩︎
Repeated roots in a characteristic equation require modifying the solution to include a term that grows linearly with n. ↩︎

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