Question

How many even numbers are greater than $202$ and less than $405$?

Answer 1

Finding number of even numbers that are greater than $202$ and less than $405$:

Step-1: Construction of an AP:

The first even number greater than $202$ is $204$ and the last even number less than $405$ is $404$.

Now, the even numbers between $202$ and $405$ form an arithmetic progression (AP) with the first term $204$ and last term $404$ and common difference $2$.

Let total number of even numbers be $n$.

Step-2: Finding the value of $n$:

Formula to be used: We know that in an AP with $n$ terms whose first term and common difference are $a$ and $d$ respectively, the $n$th term will be

$t_n=a+(n-1)d$.

Now, for our AP, $a=204$, $d=2$ and $t_n=404$. So, using the above formula we get:

$$\begin{gathered} t_n=a+(n-1)d \\ \Rightarrow 404=204+\left(n-1\right)2 \\ \Rightarrow 2(n-1)=200 \\ \Rightarrow n-1=\frac{200}{2} \\ \therefore n=101 \end{gathered}$$

Therefore, there are $101$ even numbers greater than $202$ and less than $405$.