Recurrence Relations

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# Recurrence Relations ## 3.1 Intuition: Defining Sequences Recursively A recurrence defines each term of a sequence in terms of previous terms. The Fibonacci numbers $F_n = F_{n-1} + F_{n-2}$ are the most famous example.

Recurrence Relations

3.1 Intuition: Defining Sequences Recursively

A recurrence defines each term of a sequence in terms of previous terms. The Fibonacci numbers

F_n = F_{n-1} + F_{n-2}

are the most famous example.

🔑 Key Insight: Many counting problems naturally yield recurrence relations.

3.2 Linear Homogeneous Recurrences

Form:

a_n = c_1 a_{n-1} + c_2 a_{n-2} + \cdots + c_k a_{n-k}

Characteristic equation:

r^k - c_1 r^{k-1} - c_2 r^{k-2} - \cdots - c_k = 0

Solution:

Distinct roots $r_1, \dots, r_k$ : $a_n = \alpha_1 r_1^n + \cdots + \alpha_k r_k^n$
Repeated root $r$ with multiplicity $m$ : $(\alpha_1 + \alpha_2 n + \cdots + \alpha_m n^{m-1}) r^n$

Example: Fibonacci

F_n = F_{n-1} + F_{n-2}

F_0 = 0

F_1 = 1

Characteristic:

r^2 - r - 1 = 0 \implies r = \frac{1 \pm \sqrt{5}}{2}

F_n = \frac{1}{\sqrt{5}}(\phi^n - (-\phi)^{-n})

where

\phi = \frac{1+\sqrt{5}}{2}

3.3 Non-Homogeneous Recurrences

Form:

a_n = c_1 a_{n-1} + \cdots + c_k a_{n-k} + f(n)

Solution = homogeneous solution + particular solution.

Example: $a_n = 2a_{n-1} + 3$ , $a_0 = 1$

Homogeneous:

a_n^{(h)} = \alpha \cdot 2^n

Particular: Try constant

c

c = 2c + 3 \implies c = -3

General:

a_n = \alpha \cdot 2^n - 3

a_0 = 1 \implies \alpha = 4

Solution:

a_n = 4 \cdot 2^n - 3 = 2^{n+2} - 3

✅ Practice Questions

Q1: Solve

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

a_0 = 1

a_1 = 2

Solution

$r^2 - 3r + 2 = 0 \implies r = 1, 2$ $a_n = \alpha \cdot 1^n + \beta \cdot 2^n = \alpha + \beta \cdot 2^n$ $a_0 = \alpha + \beta = 1$ , $a_1 = \alpha + 2\beta = 2 \implies \beta = 1, \alpha = 0$ $a_n = 2^n$ Q2: Solve $a_n = 4a_{n-1} - 4a_{n-2}$ , $a_0 = 0$ , $a_1 = 2$ . Solution

$r^2 - 4r + 4 = 0 \implies (r-2)^2 = 0$ , repeated root $r=2$ . $a_n = (\alpha + \beta n) \cdot 2^n$ $a_0 = \alpha = 0$ , $a_1 = (0 + \beta) \cdot 2 = 2 \implies \beta = 1$ $a_n = n \cdot 2^n$ Join Discord PreviousPigeonhole & Inclusion-ExclusionNextRecurrence Applications

What this note is really saying

Recurrence Relations

3.1 Intuition: Defining Sequences Recursively

3.2 Linear Homogeneous Recurrences

Example: Fibonacci

3.3 Non-Homogeneous Recurrences

Example: an=2an−1+3a_n = 2a_{n-1} + 3an​=2an−1​+3, a0=1a_0 = 1a0​=1

✅ Practice Questions

Example: $a_n = 2a_{n-1} + 3$ , $a_0 = 1$