Question

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

Answer 1

Step 1: Find the number of terms.

The series is $101,103,105,\dots ,199.$

Let the number of terms be$n$.

$$\begin{gathered} \Rightarrow a=101, \\ \Rightarrow d=103–101=2, \\ \Rightarrow a_n=199 \end{gathered}$$

Using the formula of $nth$term of an A.P, we get

$$\begin{gathered} a_n=a+(n–1)d \\ \Rightarrow 199=101+(n–1)2 \\ \Rightarrow 199=101+2n–2 \\ \Rightarrow 2n=199–101+2 \\ \Rightarrow 2n=100 \\ \Rightarrow n=\frac{100}{2} \\ \Rightarrow n=50 \end{gathered}$$

Step 2: Find the sum of series

To find the sum of $n$ terms

$$\begin{gathered} S_n=\frac{n}{2}[a+a_n] \\ \Rightarrow S_n=\frac{50}{2}[101+199] \\ \Rightarrow S_n=25\times \left[300\right] \\ \Rightarrow S_n=7500 \end{gathered}$$

Hence, the sum of all odd numbers between $100$ and $200$ is $7500$.