1. The problem is to find the sum of only the odd numbers in a given sequence or range. 2. To find the sum of odd numbers, we use the formula for the sum of an arithmetic series: $$S_n = \frac{n}{2} (a_1 + a_n)$$ where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term. 3. For odd numbers starting from 1, the sequence is 1, 3, 5, ..., and the $n$th odd number is given by $a_n = 2n - 1$. 4. The sum of the first $n$ odd numbers is known to be $$S_n = n^2$$. 5. This means if you want to sum the first $n$ odd numbers, you simply square $n$. 6. For example, the sum of the first 5 odd numbers (1, 3, 5, 7, 9) is $5^2 = 25$. 7. This formula works because the odd numbers form a perfect square pattern when summed. 8. Therefore, the sum of odd numbers up to the $n$th term is $$\boxed{n^2}$$.