Recursive Function Formula Steps to Solve and Worked Examples
Recursion is a process of defining objects based on previously defined other objects of the same type. A recursion relation defines some rules and a few initial values to build up an entire class of objects. Here it must be noted that if an object is defined in terms of itself, it causes self-recursion and leads to infinite nesting. A real-world example of recursion is when you are climbing a ladder. To reach the 3rd rung of the ladder, you need to reach the 2nd rung. Basically, it means that completing each step is dependent on the completion of the previous rung.
Recursive Function Definition – When a function calls itself and uses its own previous terms to define its subsequent terms, it is called a recursive function. It is the technical recursive function’s definition, i.e., a recursive function builds on itself.
The Recursive Function has 2 parts:
The value of the smallest or the first term in the sequence, usually given as f(0) or f(1)
The pattern or the rule which can be used to get the value of any term, given the value of the term preceding it. In other words, the definition of f(n) when values of f(n-1), f(n-2), etc are given.
We will now explore this by looking at the recursive function example below:
We are given a sequence of numbers 3, 5, 7, 9…
a(1) = 3 –> the first term in the series
a(n) = a(n-1) + 2 -> The rule or pattern where you need to add 2 to the last term to get the next term in the series.
So in order to find say the 6th term, we would need to find all the terms before that as shown below:
A(1) = 3
A(2) =A (1) + 2 = 3 + 2 = 5
A(3) =A (2) + 2 = 5 + 2 = 7
A(4) =A (3) + 2 = 7 + 2 = 9
A(5) =A (4) + 2 = 9 + 2 = 11
A(6) =A (5) + 2 = 11 + 2 = 13
What makes this function recursive is that it needs to know its own terms to figure out its next terms.
Expanding the recursive function formula for Arithmetic Progression – The process of defining a recursive formula for an arithmetic progression can be done by carrying below
mentioned steps:
You must determine that it is an arithmetic sequence, which means you either add or subtract the same constant value from one term to get the next term.
Find the number that you add or subtract, or the common difference between consecutive terms
Now the recursive formula can be created by stating
The first term
The formula which involves the previous term and the common difference.
So it looks like this:
a1 -> first term of the series
an = an-1 + d,
here an-1 is the previous term, d is the common difference, an is the nth term in the series, and n the ordinal number of the term
The recursive formula for a geometric sequence – It is easier to create recursive formulas for most geometric sequences than an explicit formula. In this, the common ratio can be seen easily and can be used to create the recursive formula quickly. Let us look at a recursive function example for geometric series:
3, 6, 12, 24…
Here we can see that the first term is a1 = 3 and an = 2*an-1
Now we will look at the method to write a recursive function for a geometric series:
You must determine that it is a geometric sequence, which means you either multiply or divide the same constant value from one term to get the next term.
Find the number that you multiply or divide by or the common ratio between consecutive terms
Now the recursive formula can be created by stating
The first term
The formula which involves the previous term and the common ratio.
So it looks like this:
a1 -> first term of the series
an = r * an-1
here an-1 is the previous term, r is the common ratio, an is the nth term in the series, and n the ordinal number of the term
Visualization of Recursive Function – A Pascal’s triangle is the most famous example of a recursive sequence. In this, you can see that each term is obtained by adding 2 other parts of the triangle. Below is a visualization of the triangle:
Conclusion - A recursive function is a function that builds by calling itself and is calculated using a starting value and a pattern or rule which gives the next value in the series. It can be applied to arithmetic as well as geometric series. A common difference is used to add or subtract for getting the next term in an arithmetic progression, and a common ratio is used to multiply or divide to get the next term in a geometric progression. These functions are widely used in coding algorithms where one needs to traverse hierarchies or find the factorial of a number.
FAQs on Recursive Functions Explained with Definition and Concepts
1. What is a recursive function in mathematics?
A recursive function is a function that defines each term using one or more previous terms of the same function. In mathematics, recursion builds complex sequences from simpler initial values.
- It includes a base case (starting value).
- It includes a recursive rule (formula using earlier terms).
- Each new term depends on previously defined terms.
2. What is the general form of a recursive formula?
The general form of a recursive formula includes a base case and a recurrence relation. It is usually written as:
a₁ = c (base case)
aₙ = f(aₙ₋₁) for n ≥ 2 (recursive rule)
- a₁ = c gives the starting value.
- aₙ = f(aₙ₋₁) defines each term using the previous term.
3. What is an example of a recursive function?
A classic example of a recursive function is the factorial function, defined as n! = n × (n−1)!.
Recursive definition:
- 0! = 1 (base case)
- n! = n × (n−1)! for n ≥ 1
4! = 4 × 3! = 4 × 3 × 2! = 4 × 3 × 2 × 1 = 24.
4. How do you solve a recursive sequence step by step?
To solve a recursive sequence, you repeatedly apply the recursive formula starting from the base case.
Example: Given a₁ = 2 and aₙ = aₙ₋₁ + 3
- a₁ = 2
- a₂ = 2 + 3 = 5
- a₃ = 5 + 3 = 8
- a₄ = 8 + 3 = 11
5. What is the difference between recursive and explicit formulas?
The main difference is that a recursive formula uses previous terms, while an explicit formula gives the nth term directly.
- Recursive: aₙ = aₙ₋₁ + 3
- Explicit: aₙ = 2 + (n−1)3
6. What is a base case in a recursive function?
A base case is the starting value that allows a recursive function to begin and prevents infinite repetition. It provides a fixed initial condition.
For example, in the Fibonacci sequence:
- F₁ = 1
- F₂ = 1
- Fₙ = Fₙ₋₁ + Fₙ₋₂
7. What is the Fibonacci sequence as a recursive function?
The Fibonacci sequence is defined recursively by adding the two previous terms. Its recursive definition is:
- F₁ = 1
- F₂ = 1
- Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3
8. How do you convert a recursive formula into an explicit formula?
To convert a recursive formula into an explicit formula, identify the pattern formed by the sequence and express the nth term directly.
Example (arithmetic sequence):
- Recursive: a₁ = 4, aₙ = aₙ₋₁ + 5
- Common difference = 5
- Explicit: aₙ = 4 + (n−1)5
9. What is a recursive relation in discrete mathematics?
A recursive relation (or recurrence relation) is an equation that defines each term of a sequence in terms of previous terms. It is widely used in discrete mathematics and algorithm analysis.
General example:
- aₙ = 2aₙ₋₁ + 1
- With initial condition a₁ = 1
10. What are common mistakes when working with recursive functions?
Common mistakes in recursive functions include forgetting the base case and misapplying the recursive rule. These errors lead to incorrect sequences or undefined results.
- Not clearly stating the initial condition.
- Using the wrong previous term (e.g., aₙ instead of aₙ₋₁).
- Arithmetic errors when expanding terms step by step.
- Confusing recursive formulas with explicit formulas.
