The average of odd numbers up to 100 is

### Concept & Formula The sum of the first $n$ odd natural numbers is exactly $n^2$. Therefore, the average of the first $n$ odd numbers is $\frac{n^2}{n}$, which simplifies perfectly to just $n$. $$Average\ of\ first\ n\ odd\ numbers = n$$ ### Step-by-Step Solution * **Given:** We need the average of all odd numbers up to 100. * **Deduction:** The series is 1, 3, 5, ..., 99. * First, determine how many odd numbers exist up to 100. Exactly half the numbers from 1 to 100 are odd, so $n = 50$. * **Calculation:** Since we are finding the average of the first 50 odd numbers, the average is exactly equal to $n$. * $Average = 50$. ### Exam Strategy & Shortcut You can also use the arithmetic progression (AP) midpoint shortcut. The first odd number is 1, and the last odd number before 100 is 99. The average is the sum of the first and last terms divided by 2. $\frac{1 + 99}{2} = \frac{100}{2} = 50$. ### Common Pitfall A frequent error is assuming $n = 100$ because the question says "up to 100," leading to an incorrect answer of 100. You must count *how many* odd numbers are actually in that range. ### Final Answer **Therefore, the correct answer is 50.**