A sequence consists of \(2022\) terms. Each term after the first term is \(1\) greater than the previous term. The sum of the \(2022\) terms is \(31\,341\).
Determine the sum of the terms in the odd-numbered positions. That is, determine the sum of every second term starting with the first term and ending with the second last term.
Note:
In solving the above problem, it may be helpful to use the fact that the sum of the first \(n\) positive integers is equal to \(\tfrac{n(n+1)}{2}\). That is, \[1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\]