Question

Let the sets $A=\left\{2 , 4 , 6,8 \ldots \right\}$ and $B=\left\{3 , 6 , 9 , 12 \ldots \right\}$ such that $n\left(A\right)=200$ and $n\left(B\right)=250.$ Find $n\left(A \cup B\right).$

Answer 1

Let, the last element of set $A$ is, $t_{200}=2+\left(200 - 1\right)2=400$
and the last element of set $B$ is $t_{250}=3+\left(250 - 1\right)3=750$
Common elements are $ \rightarrow 6,12,18\ldots $
We have to find the common elements upto $400$
$t_{n}=6+\left( n - 1\right)6\Rightarrow 400=6+6n-6$ $\Rightarrow n=\frac{400}{6}=66.6\ldots $
$t_{66}=6+\left(66 - 1\right)6=396$
Number of common elements $=66$
$n\left(A \cup B\right)=n\left(A\right)+n\left(B\right)-n\left(A \cap B\right)$
$=200+250-66=384$