--- type: theoretical --- 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 1. ***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: $$ a_n = 3 + (n-1) \cdot 4 $$ 2. ***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. --- ### 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: 1. **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. 2. **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](Mathematical%20Proofs%20(Induction).md). ## Footnotes [^1]: Linear homogeneous recurrence relations are equations where each term is a combination of earlier terms, with no added constants. [^2]: Repeated roots in a characteristic equation require modifying the solution to include a term that grows linearly with $n$.