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I came across this definition of recursive series in a textbook. It goes as follows: $$a_0 = 0$$ For all $n \in N$, $$a_{n+1} = 2a_{n}+n$$

I am not sure if this is just me being silly, but I somehow find the $a_{1}$ term to be awkward to fit into this definition since $n$ belongs to the set of $\{1,2,3,4....\}$. To get the $a_1$ term, I have to do: $$a_{0+1} = 2a_{0} + 0$$, but that formula says $n \in N$. Does my confusion make sense?

Sigsevduh
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  • This recursive formula allows you to compute a given term in the sequence from the previous one. For all $n\in \mathbb{N}$, you have that $a_{n+1} = 2 a_n+n$... You have already obtained $a_1= 2 a_0 + 0 = 0$, what's keeping you from continuing? For instance, $a_2=2 a_1 + 2 = 2\times 0 + 2 =2$. – PierreCarre Nov 23 '21 at 14:44

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Your confusion does make sense, but so does the equation.

Depending on where you are and what academic field you're studying in, the set of natural numbers may or may not include 0 (here's a link to a wikipedia article with an ISO standard stating 0 is included: https://en.wikipedia.org/wiki/ISO_31-11). This is a matter of lack of standardization of definition.

ghost
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  • To add to this answer, generally, in mathematics, we define $\Bbb N$ as the set of natural numbers including 0, and $\Bbb N^$ as the set of natural numbers excluding 0. The notation, in the more general case, tends to mean "the same set, but with the non-invertible elements removed". In $\Bbb N$, only 0 is non-invertible (meaning you can't divide by 0), so $\Bbb N^*$ only removes 0 from $\Bbb N$. – Tristan Duquesne Nov 23 '21 at 15:21