<s>[INST] Denoting the sum of $n$ odd numbers $1+3+5+\cdots+(2n-1) = 1+3+5+\cdots+(2n+1)$? Let's say we are denoting the sum of $n$ odd numbers. 
Then in symbols $1+3+5+\cdots+(2n-1)$.
If we substitute $(k+1)$ for $n$. $2n-1=2k+1$ 
So $1+3+5+...+2k+1$
Then can we use  $1+3+5+\cdots+(2n+1)$ instead of $1+3+5+\cdots+(2n-1)$?
Logically, I think they are the same, but when I think of the number of terms, $1+3+5+\cdots+(2n+1)$ has one more term than $1+3+5+\cdots+(2n-1)$.
I also think $1+3+5+\cdots+(2n+1)$ is equal to $1+3+5+\cdots+(2n-1)+(2n+1)$, so it would be contradiction. But I don't know how to explain $1+3+5+\cdots+(2n-1)=1+3+5+\cdots+2k+1$ is a contradiction.
 [/INST] If you're writing $1+3+5+\cdots+45$, for instance, you could describe that as
$1 + 3 + 5 + \cdots + (2n-1)$ for $1 \le n \le 23$, and there are 23 terms.
You could also describe it as
$1 + 3 + 5 + \cdots + (2n+1)$, but this time we have $0 \le n \le 22$, and there are still only 23 terms: even though the last value of $n$ is 22, this time we're also counting $n=0$.
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