Find the sum of the odd numbers between 0 and 50.
\(625\)
Step 1: Identify the sequence of odd numbers.
The odd numbers between 0 and 50 are:
\[1, 3, 5, 7, \ldots, 49\]
This is an arithmetic progression (AP) with:
First term \(a = 1\), common difference \(d = 2\), last term \(l = 49\).
Step 2: Find the number of terms.
Use the nth-term formula:
\[a_n = a + (n - 1)d\]
Put \(a_n = 49\):
\[49 = 1 + (n - 1)2\]
\[49 - 1 = 2(n - 1)\]
\[48 = 2(n - 1)\]
\[n - 1 = 24 \Rightarrow n = 25\]
So, there are 25 odd numbers.
Step 3: Use the sum formula for an AP.
\[S_n = \dfrac{n}{2}(a + l)\]
Substitute \(n = 25\), \(a = 1\), \(l = 49\):
\[S_{25} = \dfrac{25}{2}(1 + 49) = \dfrac{25}{2} \cdot 50 = 25 \cdot 25 = 625\]
Conclusion: The sum of all odd numbers between 0 and 50 is 625.