My Math Forum Recursive definition and induction

 Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion

 December 1st, 2018, 12:29 PM #1 Newbie   Joined: Nov 2018 From: France Posts: 8 Thanks: 0 Recursive definition and induction 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.
 December 1st, 2018, 01:01 PM #2 Global Moderator   Joined: May 2007 Posts: 6,821 Thanks: 723 Statement is not clear. Every number is $a_1+a_1+a_1+.....$ Thanks from greg1313
December 1st, 2018, 01:09 PM   #3
Newbie

Joined: Nov 2018
From: France

Posts: 8
Thanks: 0

Quote:
 Originally Posted by mathman Statement is not clear. Every number is $a_1+a_1+a_1+.....$
as a sum of different elements of the series...
Every natural number can be written from these ones ^

That's what we have to prove.

December 2nd, 2018, 07:25 PM   #4
Senior Member

Joined: Sep 2016
From: USA

Posts: 642
Thanks: 406

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
 Originally Posted by 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.

Write out more terms from this sequence and its obvious.

 Tags definition, induction, recursive

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Sylvia Pen Applied Math 14 June 7th, 2013 04:09 PM stranger Applied Math 12 February 5th, 2012 09:20 AM philip Algebra 4 December 27th, 2011 11:23 AM dwnielsen Real Analysis 0 December 19th, 2009 09:41 AM robocop_911 Applied Math 3 June 1st, 2008 09:43 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top