Notation for summation while skipping elements

Question

Suppose I have a summation like so:

$\sum_{i =0}^n l^i$

Except I don't want to compute for all $0 \leq i \leq n$. I just want to compute it for the arithmetic sequence: $1, 3, 5, 7, 9...$ How do I write this in the summation notation?

Answer 1

Use $2i+1$ instead of $i$ in the expression to be summed.$$\sum_{i=0}^k l^{2i+1}$$ Where $2k+1=n$

Answer 2

Like this: $$\sum^n_{i=0}_\text{i is odd} \text{or just} \sum^n_{i=0}_\text{odd}$$ And in general you'd just replace "i is odd" with whatever criteria you have.

Or you could mention it outside the sum. As in, immediately after the sum write "where $i$ is odd" or "where $i \in \{x \in \Bbb N\ |\ x = 2k \wedge k \in \Bbb N\}$".