# Recursive definition and induction

• MHB
• CStudent
In summary, the series $a_n$ is defined by a recursive formula $a_n = a_{n-1} + a_{n-3}$. Every natural number can be written as a sum (of one or more) of different elements of the series. However, proving this is not an easy task, as there is a problem with powers of 3.
CStudent
Hey.

The series $a_n$ is defined by a recursive formula $a_n = a_{n-1} + a_{n-3}$ and its base case is $a_1 = 1 \ a_2 = 2 \ a_3 = 3$.
Prove that every natural number can be written as a sum (of one or more) of different elements of the series $a_n$.

Now, I know that is correct intuitively but I don't know how to prove that.
Generally, I have some problem of understanding the concept of recursion.

Thanks.

CStudent said:
Hey.

The series $a_n$ is defined by a recursive formula $a_n = a_{n-1} + a_{n-3}$ and its base case is $a_1 = 1 \ a_2 = 2 \ a_3 = 3$.
Prove that every natural number can be written as a sum (of one or more) of different elements of the series $a_n$.

Now, I know that is correct intuitively but I don't know how to prove that.
Generally, I have some problem of understanding the concept of recursion.
Since you're unclear on the concept of recursion, it would be a good idea to calculate a few terms of the sequence. (Note that what you have here is a sequence of numbers. A series is the sum of numbers in some sequence.)

From the given info we have:
##a_1 = 1##
##a_2 = 2##
##a_3 = 3##
Using the recursion formula, we get:
##a_4 = a_3 + a_1 = 3 + 1 = 4##
##a_5 = a_4 + a_2 = 4 + 2 = 6##
##a_6 = a_5 + a_3 = 6 + 3 = 9##
and so on...

CStudent said:
Hey.

The series $a_n$ is defined by a recursive formula $a_n = a_{n-1} + a_{n-3}$ and its base case is $a_1 = 1 \ a_2 = 2 \ a_3 = 3$.
Prove that every natural number can be written as a sum (of one or more) of different elements of the series $a_n$.

Now, I know that is correct intuitively but I don't know how to prove that.
Generally, I have some problem of understanding the concept of recursion.

Thanks.
It's not a really easy proof, I think.

- How would you go about writing a natural number as a sum of the elements of a sequence. I think there's a rather obvious algorithm.
- What can go wrong with this algorithm. Why does it fail with powers of 3.
- How can you prove that this can't happen here. (using the recursion)

Greg Bernhardt

## 1. What is a recursive definition?

A recursive definition is a definition that defines a term or concept in terms of itself. This means that the definition refers back to itself, creating a loop. Recursive definitions are often used in mathematics and computer science to define algorithms and functions.

## 2. How does recursion work in computer programming?

In computer programming, recursion is a technique where a function calls itself within its own code. This allows for a problem to be solved by breaking it down into smaller, simpler versions of itself. Recursion can be a powerful tool for solving complex problems, but it can also lead to infinite loops if not implemented correctly.

## 3. What is the principle of mathematical induction?

The principle of mathematical induction is a proof technique used to prove statements about natural numbers. It involves proving a base case, and then showing that if the statement is true for some number, it is also true for the next number. This process is repeated until the statement is proven to be true for all natural numbers.

## 4. How is recursion related to induction?

Recursion and induction are related in that they both involve breaking down complex problems into smaller, simpler versions of themselves. In recursion, a function calls itself to solve a problem. In induction, a statement is proven to be true for all natural numbers by showing it is true for a base case and then for the next number using the principle of mathematical induction.

## 5. What are some real-life examples of recursion?

Some real-life examples of recursion include the process of making a peanut butter and jelly sandwich, where you repeat the same steps for each layer of the sandwich, and the concept of fractals, where a shape is made up of smaller versions of itself. Other examples include the Fibonacci sequence, the Tower of Hanoi puzzle, and certain sorting algorithms.

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