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 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: 6,761 Thanks: 696 Statement is not clear. Every number is $a_1+a_1+a_1+.....$ Thanks from greg1313
December 1st, 2018, 01:09 PM   #3
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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
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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.

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