Question

find the some of all natural numbers lying between 100 to 200 which are divisible by 4?

Answer 1

The list of numbers between 100 and 200 divisible by 4 are :
104, 108, ..........196.
The above list is in AP with a = 104 and d = 4
Let a_n = 196
$$\begin{gathered} \Rightarrow a+(n-1)d=196 \\ \Rightarrow 104+(n-1)4=196 \\ \Rightarrow (n-1)4=92 \\ \Rightarrow n-1=23 \\ \Rightarrow n=24 \end{gathered}$$

$$\begin{gathered} S_{24}=\frac{24}{2}\left[2\left(104\right)+23\left(4\right)\right] \\ =12\left[208+92\right] \\ =12\times 300=3600 \end{gathered}$$