Q66 / 170 · Manipal (MET) 2016 · Mathematics

Total number of regions in which ' $n$ ' coplanar lines can divide the plane, it is known that no two lines are parallel and no three of them are concurrent, is equal to

  • Correct answer
    A
    $\frac{1}{2}\left(n^2+n+2\right)$
  • B
    $\frac{1}{2}\left(n+3 n^2\right)$
  • C
    $\frac{1}{2}\left(3 n+n^2\right)$
  • D
    $\left(n^2-n+2\right)$

Solution

Let number of regions for $n$ lines be $R(n)$. Clearly, $R(1)=2, R(2)=4, \ldots$ $R(n)=R(n-1)+n \text { i.e. } R(n)-R(n-1)=n$ Putting $n=2,3, \ldots, n$ and then adding, we get $R(n)-R(1)=2+3+\ldots+n$ $\begin{aligned} & =2+2+3+4+\ldots+n=1+\frac{n(n+1)}{2} \\ & =\frac{n^2+n+2}{2}\end{aligned}$

Hence, the correct answer is option: 'A'.

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