Step-by-Step Solution
Key Concept: The odd numbers between 0 and 50 form an arithmetic progression (AP) with first term $a=1$, common difference $d=2$, and last term $l=49$. Use the AP sum formula $S_n = \frac{n}{2}(a+l)$ after determining the number of terms $n$.
1. Identify the AP\
The odd numbers are $1, 3, 5, \dots , 49$.\
Hence, $a = 1$, $d = 2$, $l = 49$.\
2. Find the number of terms ($n$)\
$$n = \frac{l - a}{d} + 1 = \frac{49 - 1}{2} + 1 = \frac{48}{2} + 1 = 24 + 1 = 25.$$\
3. Apply the sum formula for an AP\
$$S_n = \frac{n}{2}\,(a + l)$$\
Substituting the values:\
$$S_{25} = \frac{25}{2}\,(1 + 49) = \frac{25}{2}\times 50 = 25 \times 25 = 625.$$\
4. State the result\
The sum of all odd numbers between 0 and 50 is $\boxed{625}$.
Correct Answer: 625