Hi

This is a standard recurrence problem.

No two lines are parallel and no 3 are concurrent.

 

                  L_n means the number of regions , n line(s) have divided into.

Observe that L₀= 1. That is no  line means one plane.

L₁=2. one line divides the plane into 2 regions. similarly

L₂=4, 

L₃=7 etc....( draw and verify!)

Observing the pattern we conclude that

L_n=L_n-1+ n        for n> 0

   = L_n-2+(n-1)+n       ( this technique is called unfolding)

   = L_n-3+(n-2)+(n-1)+1

   = .........................

   = L₀+1+2+3+.............+(n-2)+(n-1)+n

   = 1 + S_n            where S_n= 1+2+.......+(n-1)+n

   =1+ n(n+1)/2     ( u get the proof in many textbooks)

 

Hence the total no. of regions 17 lines divide are

  17 * (17+1)  /2 +1

  154.

 

Cheers!