Question

A line divides a plane into 2 regions. Two lines divide the plane into maximum 4 regions. If $L_n$ is the maximum number of regions divided by $n$ lines then the following is/are true?

• A. $L_{20}=211$
• B. $L_{10}=56$
• C. $L_{15}=121$
• D. $L_{25}=326$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $L_{20}=211$
B $L_{10}=56$
C $L_{15}=121$
D $L_{25}=326$




$L_1=2,L_2=4$
$L_3=7$

When we introduce $n^{th}$ line, no. of plane increases by $n$.
$L_n=L_{n-1}+n$
So $L_n-L_{n-1}=n$
$L_{n-1}-L_{n-2}=n-1$
...
...
...
$L_2-L_1=2$

Adding all
$L_n=\frac{n(n+1)}{2}+1$
$L_{10}=56$
$L_{15}=121,L_{20}=211,L_{25}=326$