Question

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

Answer 1

All the odd numbers between 100 and 200 are:
101, 103...199
Here, we have:
$$\begin{gathered} a=101 \\ d=2 \\ {a}_n=199 \\ \Rightarrow 101+(n-1)\times 2=199 \\ \Rightarrow 2n-2=98 \\ \Rightarrow 2n=100 \\ \Rightarrow n=50 \\ {S}_n=\frac{n}{2}\left[2a+(n-1)d\right] \\ \Rightarrow S_{50}=\frac{50}{2}\left[2\times 101+(50-1)2\right] \\ \Rightarrow S_{50}=25\left[202+98\right] \end{gathered}$$
$\Rightarrow S_{50}=7500$