Question

If n lines are drawn on a plane, no two lines are parallel, and all lines intersect at different points, how many sections do they separate the plane into? assume that no more than two lines intersect at any one point and prove your result using weak induction. Don’t include unneeded base cases.

Answer 1

Proved our result using weak induction.

Given:

If n lines are drawn on a plane, no two lines are parallel, and all lines intersect at different points, how many sections do they separate the plane into, assume that no more than two lines intersect at any one point.

Suppose it is true for n .Now add a new line , the number of  vertices increases by n one with each of the old line and the number of line segments increases by 2n+1 ( Each of the n old lines is broken into one more piece and the new line has n+1 segments , hence by euler's theorem on planar graphs the number of regions (F) increases by n+1 since V - E + F is constant and V - E is reduced by n+1.

so we now have

$n^{2}$+n+2/2 + n + 1 = $(n+1)^{2}$+ n+1+2/2  vertices are desired

Therefore proved our result using weak induction.

Learn more about the weak induction and lines here:

brainly.com/question/14642442

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