Recurrence relations, also known as recursive equations, are equations that recursively define sequences or functions in terms of their previous values. They are commonly used in mathematics, computer science, and various other fields to describe processes or systems that evolve over time or in stages.

In a recurrence relation, the value of a term an​ in a sequence or function is expressed in terms of one or more preceding terms. The initial conditions or base cases define the starting values of the sequence.

A typical form of a recurrence relation is:

an​=f(an−1​,an−2​,…,an−k​)

Where:

• an​ is the
  $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">n</span></span></span></span>-th\; term\; in\; the\; sequence.</span>}$

• $f$ is a function that determines the
  $\mathrm{<br>}$

  n-th term based on the previous 

  
  $$

  k terms (where
  $$

  k is a constant).

Example:

The Fibonacci sequence is a classic example of a sequence defined by a recurrence relation:

Fn​=Fn−1​+Fn−2​

with initial conditions
${}_{}$

F0​=0 and
${}_{}$

F1​=1.

Growth of Functions

When analyzing algorithms or processes, it’s essential to understand the growth rate of functions. Recurrence relations often arise in the analysis of algorithms, particularly in the context of time complexity.

In the analysis of algorithms, the growth rate of a function describes how its value increases as the input size grows. Common growth rates include linear, logarithmic, polynomial, exponential, and factorial.

Recurrence relations provide a way to express the time complexity of algorithms recursively. Solving recurrence relations helps determine the asymptotic behavior of algorithms and understand their efficiency.

Example:

Consider the recurrence relation describing the time complexity of the merge sort algorithm:

T(n)=2T(n/2)+n

This recurrence relation states that the time taken to sort an array of size
$$

n is twice the time taken to sort two arrays of size
$$

n/2, plus

n for the merge step. Solving this recurrence relation reveals that the time complexity of merge sort is
$$

O(nlogn), indicating a logarithmic growth rate.

In summary, recurrence relations are valuable tools for describing sequences and functions that evolve over time, while the analysis of their solutions provides insights into the growth rates of functions, particularly in the context of algorithmic time complexity.