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December 1st, 2018, 01:29 PM   #1
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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.
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December 1st, 2018, 02:01 PM   #2
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Statement is not clear. Every number is $a_1+a_1+a_1+.....$
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December 1st, 2018, 02:09 PM   #3
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Quote:
Originally Posted by mathman View Post
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.
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December 2nd, 2018, 08:25 PM   #4
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Quote:
Originally Posted by CStudent View Post
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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