Find the sum of all odd numbers between 100 and 200.

All the odd numbers between 100 and 200 are:

101, 103...199

Here, we have:

$a=101$

$d=2$

$a_{n}=199$

$\Rightarrow 101+(n-1) \times 2=199$

$\Rightarrow 2 n-2=98$

$\Rightarrow 2 n=100$

$\Rightarrow n=50$

$S_{n}=\frac{n}{2}[2 a+(n-1) d]$

$\Rightarrow S_{50}=\frac{50}{2}[2 \times 101+(50-1) 2]$

$\Rightarrow S_{50}=25[202+98]$

$\Rightarrow S_{50}=7500$