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4. Suppose there are n lines in the plane, no two are parallel and no three are intersecting at one point. Into how many regions is the plane divided by these lines? Let n) denote the number of regions the plane is divided into by 11 lines. Do the following. [a] Compute the values for n} for n. = 1.2.3.45. {In} Give a conjecture on the formula of n). to} Justify the conjecture using induction. 3! = 3.2.1 Proof by Induction . h Conjecture 2' = 2 for all integers r nz4 (in ! > 2 " 2 2. = 2 2 = 4 Proof You show that * Is 3. = 4 12 = 32 6 n! = 4! = 4.3. 2.1 = 24 6! = 720 > 2) = 64 10 24 = 2.2.2.2 = 16 . 7 7! = 3,040 2 2+ = 128 (?)8 40, 328 X 2521 thus , $ is true for 1= 4 : 8! that is, n! > 2" for n= 4. Assume that for neke. . Induction Step . Then we want to show that * istrue for he Re+ 1 . . Weknow that 7!727. (9 ) 8 =23 Let $ 24 be given us Induct D To show : 8! 7 28 ) joint 1=8 suppose that () is the for Hypha 7! n= R . ( IH ) 8! = 8 7.6. 5.4. 3.2.1 6 That is, k ! > 2. for =8 7!12 8 27by () Coup inlage GOAL 2. 2 = 2 1 28 we want to show that Thus , 8 ! > 28 ( *+ 1 ) ! > 2 Rty let/ Conclusion = 2', 2" Now , By the principle of ( R+ 1 )! = ( 1 + 1 ) .R ! Show in duction , it follows R+1 72 ( KID ). 2 by J.H . that Since R24) whee have 2 . (2. is the for 42+ 1 7 4+11=572. 121 1 ALL all integg Thus, 1+172 if *zd. Thus, (*) holds for 1= R+1 , n2d. completing the induction stef