Set Notation Review. N the set of positive integers (aka set of natural numbers) {1, 2, 3, }

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1 11. Notes o Mathematical Iductio Before we delve ito the today s topic, let s review some basic set otatio Set Notatio Review N the set of positive itegers (aa set of atural umbers) {1,, 3, } Z the set of itegers {, -3, -, -1, 0, 1,, 3, } Q the set of ratioal umbers (writte as a fractio a/b) R the set of real umbers Notice how all sets are deoted by CAPITAL LETTERS. This is because lower case letters were already reserved for use by variables. Cool fact, huh? May importat results i mathematics have bee discovered by observig patters i some specific cases ad the by maig geeralizatios from the observatios. This is called Applyig INDUCTIVE REASONING. Mathematicias would the use their deductive reasoig sills to come up with rules i fact, they were called cojectured rules (pretty facy) Now, mathematicias would give example after example provig that their cojectured rule is TRUE. Nice as this is, your frieds oly eed oe couterexample to prove that your cojecture is NOT TRUE. Let s see 3 is prime, so is 5, ad 7, ad 11 I cojecture that all prime umbers are odd. Wow, my first cojecture I woder what my math frieds will thi? Let s see Hey guys! Guess what? Hey Marvi. Hi I came up with a cojecture O, let s hear it Time to impress All prime umbers are odd! What about? Yeah. is prime ad is ot odd! Aw shux!! That s o, you re still cool! Iductio Priciple: mathematicias use this idea to prove properties with Natural umbers. It wors lie this Da Muscarella, 00

2 THE PRINCIPAL OF MATHEMATICAL INDUCTION Let P be a statemet ivolvig the positive iteger. If 1. P 1 is true for the atural umber 1 (some boos write this as P(1), or simply 1) AND. The truth of P implies the truth of P +1, for every positive, the P must be true for all positive itegers. BOTH PARTS ARE NECESSARY TO USE THE PRINCIPAL OF MATHEMATICAL INDUCTION Now, there are two sides to every equatio ad this is o differet. S Term formula + ( 1) Sum formula The left side is called the term formula side, ad the right is called the sum formula side. Did you ow that the first term does t always have to be 1? Nope I did t Really? Usually it is, but it ca be other umbers, too! Wow Cool What, lie? You re soooo fuy! Da Muscarella, 00

3 Speaig of terms, let s review what the ext term is for give situatios where represets a iteger Term Next term Simplified versio of ext term ( + 1) ( + 1) ( + 1) ( + 1) ( + 1) ( + 1) ( + 1) 1 ( + 1) ( + 1)(( + 1) + 1) ( + 1)( + ), or ( + 1) For geometric proofs, you eeded Statemets ad Reasos. Mathematical Iductio proofs eed four parts: Whe 1, Assume P() is true for 1, The, ad Therefore. Da Muscarella, 00

4 EX 1] Use mathematical iductio to prove the statemet is true for all atural umbers. + ( 1) S Idea whe we get to the ed of the THEN part we wat the right side to be i the form. Sometimes it is easier to wor bacwards if you get stuc. ( + 1)( + ) Whe 1 1 (1 + 1) 1 1() 1 () Simplify 1 1 (3) P(1) is true! (1) Write 1 o the left (sice it is the first term), ad substitute 1 for o the sum formula (right) side If the first term were 7, we d put 7 o the left. Assume P() is true for 1. () You have to actually state this ( + 1) S (5) Replace with mathematicias are soooo picy about how they write thigs! The, ( + 1) S ( + 1) + ( + 1) () Chage S to S +1,add ( + 1) to both sides ( + 1) ( + 1) +1 (7) Remember Factor By Groupig? From (7) oward, you ( + 1) ( + 1) + (8) Rewrite 1 as wor with the right side ( + 1) ( + 1) (9) Simplify ( + 1) ( + 1) ( + ) (10) Simplify Therefore, ( 1) is true for all N. (11) State coclusio Da Muscarella, 00

5 EX ] For this problem, we will loo at two differet approaches. Use mathematical iductio to prove P ( + 1) is true for all atural umbers. Whe 1 1(1+1) (1) Write o the left ad 1 for o the sum formula (right) side 1() () Simplify (3) P(1) is true! Assume P() is true for 1. () You have to actually state this The, S ( + 1) (5) Replace with mathematicias are soooo picy about how they write thigs! S ( + 1) ( + 1) + ( + 1) () S + ( + 1) ( + 1)( + ) (7) You ca substitute S for o the left ad use Factor By Groupig o the right Therefore, ( + 1) is true for all N. (8) State coclusio Example ] S ( + 1) Alterate Techique Whe 1 1(1 + 1) (1) Write o the left ad 1 for o the sum formula (right) side 1() () Simplify (3) P(1) is true! Assume P() is true for 1. S ( + 1) () You have to actually state this (5) Replace with The, S ( + 1) ( + 1) + ( + 1) () S + ( + 1) (7) Substitute S ad the distribute o the right side S + ( + 1) (8) Simplify S + ( + 1) ( + 1)( + ) (9) Factor right side ad we re doe!! Therefore, S ( + 1)is true for all N. (10) State coclusio Da Muscarella, 00

6 EX 3] Use mathematical iductio to prove P ( 1) ( + 1) is true for all atural umbers. Idea whe we get to the ed of the THEN part we wat the right side to be i the form ( +1)(( + 1) + 1) or ( +1)( + 3 ) Whe 1 3 1((1)+1) (1) Write 3 o the left ad 1 for o the sum formula (right) side 3 1(3) () Simplify 3 3 (3) P(1) is true! Assume P() is true for 1. () You have to actually state this S ( 1) ( + 1) (5) Replace with The, S ( 1) + ((+1) 1) ( + 1) + ((+1) 1) () S + (+3) ( + 1) + ( + 3) (7) Substitute S ad the simplify ((+1) 1) to ( + 3) S + (+3) (8) S + (+3) (9) Simplify S + (+3) ( +1)( + 3) (10) Factor S + (+3) ( +1)( + + 1) (11) Split 3 ito + 1 S + (+3) ( +1)(( + 1) + 1) (1) Therefore, ( 1) ( + 1) is true for all N. (13) State coclusio Da Muscarella, 00

7 Sums of Powers of Itegers: + ( 1) ( + 1)( + 1) ( + 1) ( + 1)( + 1)( ( + 1) ( EX ] Fid the sum usig the formulas for the sums of powers of itegers for S 5 1 Solutio: First, determie how may terms are i the sum 1,, 3,, 5 We have 5 terms S 5 1 ( + 1)( + 1)(3 + 3 (1) Write sum formula for 5((5) + 1)( ( 5) + 1)(3(5) + 3( 5) () Substitute 5 for 5()(10 + 1)( (3) Simplify (11)(89) () Simplify 979 (5) Simplify *Verify o your graphig calculator. Da Muscarella, 00

8 Math Aalysis Name: 11.5 Notes o Mathematical Iductio Bloc: Date: For geometric proofs, you eeded Statemets ad Reasos. Mathematical Iductio proofs eed four parts: Use mathematical iductio to prove the give statemet is true for all atural umbers. + ( 1) EX 1] S EX ] S ( + 1) EX 3] S ( 1) ( + 1) EX ] Usig the Sums of Powers of Itegers formula, fid the sum of 5. 1 Sums of Powers of Itegers + ( 1) ( + 1)( + 1) ( + 1) ( + 1)( + 1)( ( + 1) ( HW Problems: Use the Pricipal of Math Iductio to prove that the give statemet is true for all atural umbers,. 1] S (3 ) ( 3 1) ] S ] S ( + 1)(+ 1) ] S (5 3) ( 5 1) Use the formulas for the sums of powers of itegers fid the sum of each. 5]. ] 1 i 1 3 i [#5 ad # HINT: use formulas!] 8i Da Muscarella, 00

9 Math Iductio HW Problems Key: 1] S (3 ) ( 3 1) Whe ( 3(1) 1 ) (1) Write 1 o the left ad 1 for o the sum formula (right) side 1 1 ( ) () Simplify 1 1 (3) P(1) is true! Assume P() is true for 1. S (3 ) ( 3 1) () You have to actually state this (5) Replace with The, S (3 ) + (3(+ 1) ) ( 3 1) S + (3(+ 1) ) ( 3 1) S + (3(+ 1) ) ( 3 1) + (3(+ 1) ) () + (3 + 3 ) (7) Substitute S ad the simplify + (3 + 1) (8) S + (3(+ 1) ) ( 3 1) + ( ) S + (3(+ 1) ) ( (3 1) (3 1) ) S + (3(+ 1) ) ( ) S + (3(+ 1) ) ( ) (9) Rewrite (10) Factor out (11) (1) 1 ( 1)(3 ) + + (13) 1 ( 1)(3 3 1) + + (1) 1 ( 1)(3( 1) 1) + + (15) ( + 1) (3( + 1) 1) (1) S + (3(+ 1) ) ( ) S + (3(+ 1) ) ( ) S + (3(+ 1) ) ( ) S + (3(+ 1) ) ( ) Therefore, (3 ) ( 3 1) is true for all N. (17) State coclusio Da Muscarella, 00

10 ] S Whe (1) Write 1 o the left ad 1 for o the sum formula (right) side 1 1 () Simplify 1 1 (3) P(1) is true! Assume P() is true for 1. () You have to actually state this S (5) Replace with The, S () S (7) Substitute S S + () 1 (8) Simplify S (9) Remember, same base for (), so add expoets Therefore, S is true for all N. (10) State coclusio Da Muscarella, 00

11 3] S ( + 1)(+ 1) Whe 1 1 1(1 + 1)((1) + 1) (1) Write 1 o the left ad 1 for o the sum formula (right) side 1 1()(3) () Simplify 1 1 (3) P(1) is true! Assume P() is true for 1. S ( + 1)(+ 1) () You have to actually state this (5) Replace with The, S ( + 1) ( + 1)(+ 1) + ( + 1) () S + ( + 1) ( + 1)(+ 1) ( + 1) + (7) Substitute S ad get commo deomiator [ ] S + ( + 1) ( + 1) (+ 1) + ( + 1) ( + 1) S + ( + 1) ( + 1) 7 S + ( + 1) + + S + ( + 1) ( + 1)( + )(+ 3) S + ( + 1) ( + 1)(( + 1) + 1) + (+ + 1) S + ( + 1) ( + 1)(( + 1) + 1) + (( + 1) + 1) Therefore, S ( + 1)(+ 1) (8) Factor out ( + 1) (9) Distribute (10) Simplify (11) Could stop here ad go to (1) (1) (13) is true for all N. (1) State coclusio Da Muscarella, 00

12 1 5(1) 1 (1) Write 1 o the left ad 1 for o the sum formula (right) side 1 ( ) () Simplify (3) P(1) is true! ] S (5 3) ( 5 1) Whe 1 ( ) Assume P() is true for 1. () You have to actually state this S (5 3) ( 5 1) (5) Replace with The, + (5( + 1) 3) () S + (5 + ) ( 5 1) + (5 + ) (7) Substitute S ad simplify (5+ ) S + (5 + ) ( 5 1) + (8) Get commo deomiator S + (5 + ) (9) S + (5 + ) (10) ( + 1)(5+ ) S + (5 + ) (11) ( + 1)(5( + 1) S + (5 + ) (1) S (5 3) + (5( + 1) 3) ( 5 1) Therefore, S (5 3) ( 5 is true for all N. (13) State coclusio Use the formulas for the sums of powers of itegers fid the sum of each. 5] 1 ( + 1)(() + 1) ( + 1) (7)(13) (7) You have to show the substitutio ito the formulas to receive credit!!! 3 ] i 8i i 1 ( + 1)(() + 1) () ( + 1) 8 (7)(13) 8(3)(9) Da Muscarella, 00

13 Still completig this part EX 5] Prove 1 ( + 1)( + 1)(3 + 3 Whe 1 1 1(1 + 1)( 1+ 1)( ()(3)(5) 1 1 (1) 1 1 (left side), substitute o right side () Simplify (3) Simplify 1 1 (3) P(1) is true! Assume S() is true for 1. () State this S ( + 1)( + 1)(3 + 3 (5) Replace with The, S ( + 1) ( + 1)( + 1)( ( + 1) () S + ( + 1) S + ( + 1) ( + 1)( + 1)(3 + 3 ( + 1)( + 1)( ( + 1) (7) Substitute + ( + 1) (8) LCD S + ( + 1) S + ( + 1) ( + 1)( + 1)( ( + 1)( + 1)( ( + 1) ( + 1) (9) Simplify (10) Simplify S + ( + 1) ( Da Muscarella, )[ ( + 1)( ( + 1) 3 ] (11) Factor By Groupig

14 S + ( + 1) ( S + ( + 1) ( 3 + 1)[ )[( )( ] 5)] (1) Simplify (13) Factor You ca tha me later S + ( + 1) ( S + ( + 1) ( + 1)[( + )( + 3)( )] + 1)( + )[(( + 1) + 1))( )] (13) Factor (1)Rewrite + 3 as ( + 1) + 1 S + ( + 1) ( + 1)( + )[(( + 1) + 1))(3( + 1) + 9( + 1) 1)] (15) I (15), we replace every with ( + 1). The, use your algebra sills to determie what umber goes here ( + 1) + 9( + 1) + 3( + + 1) Da Muscarella, 00