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Mathematical Induction Mathematical Induction Jason Filippou CMSC250 @ UMCP 06-27-2016 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 1 / 48 Outline 1 Sequences and series Sequences Series and partial sums 2 Weak Induction Intro to Induction Practice 3 Strong Induction 4 Errors in proofs by mathematical induction Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 2 / 48 Sequences and series Sequences and series Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 3 / 48 Sequences and series Sequences Sequences Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 4 / 48 Examples: 2; 4; 6;::: 10; 20; 30 1; 1; 2; 3; 5; 8; 13; 21;::: So, sequences can be either finite or infinite. We will mostly care about infinite sequences. Sequences and series Sequences Definitions Definition (Sequence) A function a : N 7! R is called a sequence. Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 5 / 48 So, sequences can be either finite or infinite. We will mostly care about infinite sequences. Sequences and series Sequences Definitions Definition (Sequence) A function a : N 7! R is called a sequence. Examples: 2; 4; 6;::: 10; 20; 30 1; 1; 2; 3; 5; 8; 13; 21;::: Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 5 / 48 Sequences and series Sequences Definitions Definition (Sequence) A function a : N 7! R is called a sequence. Examples: 2; 4; 6;::: 10; 20; 30 1; 1; 2; 3; 5; 8; 13; 21;::: So, sequences can be either finite or infinite. We will mostly care about infinite sequences. Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 5 / 48 Described through an explicit formula... k bk = 2 rn = (n + 1)! Or a recursive formula... Fn+1 = Fn + Fn−1 8n ≥ 1 Sequences and series Sequences Denoting sequences A sequence can be enumerated... a : a1; a2;::: (or just a1; a2;::: ) c0; c1; c2;::: (notice the indices) Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 6 / 48 Or a recursive formula... Fn+1 = Fn + Fn−1 8n ≥ 1 Sequences and series Sequences Denoting sequences A sequence can be enumerated... a : a1; a2;::: (or just a1; a2;::: ) c0; c1; c2;::: (notice the indices) Described through an explicit formula... k bk = 2 rn = (n + 1)! Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 6 / 48 Sequences and series Sequences Denoting sequences A sequence can be enumerated... a : a1; a2;::: (or just a1; a2;::: ) c0; c1; c2;::: (notice the indices) Described through an explicit formula... k bk = 2 rn = (n + 1)! Or a recursive formula... Fn+1 = Fn + Fn−1 8n ≥ 1 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 6 / 48 a0 and ! fully define the sequence. So, how can I write ar? a1 + r ∗ ! r ∗ a0 a0 + r ∗ ! r ∗ a0 Sequences and series Sequences The arithmetic sequence Definition ∗ Let a : a0; a1;::: be a sequence and ! 2 R: If aj = aj−1 + ! 8j 2 N , a is an arithmetic sequence (or progression). Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 7 / 48 Sequences and series Sequences The arithmetic sequence Definition ∗ Let a : a0; a1;::: be a sequence and ! 2 R: If aj = aj−1 + ! 8j 2 N , a is an arithmetic sequence (or progression). a0 and ! fully define the sequence. So, how can I write ar? a1 + r ∗ ! r ∗ a0 a0 + r ∗ ! r ∗ a0 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 7 / 48 a0 and c fully define the sequence. ar. How can I write it? r r rc r c ∗ a0 a0 a0 a0 + c Sequences and series Sequences The geometric sequence Definition ∗ ∗ Let a : a0; a1;::: be a sequence and k 2 R : If aj = c ∗ aj−1 8j 2 N , a is a geometric sequence (or progression). Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 8 / 48 Sequences and series Sequences The geometric sequence Definition ∗ ∗ Let a : a0; a1;::: be a sequence and k 2 R : If aj = c ∗ aj−1 8j 2 N , a is a geometric sequence (or progression). a0 and c fully define the sequence. ar. How can I write it? r r rc r c ∗ a0 a0 a0 a0 + c Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 8 / 48 Sequences and series Series and partial sums Series and partial sums Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 9 / 48 Definition (Partial sum) +1 X Let n 2 N. Then, the n-th partial sum of the series ai, denoted i=0 n X Sn, is the sum ai. i=0 The partial sums themselves also form a sequence! Sequences and series Series and partial sums Definitions Definition (Series) +1 X Let a0; a1;::: be any sequence. Then, the sum ai is called a series. i=0 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 10 / 48 The partial sums themselves also form a sequence! Sequences and series Series and partial sums Definitions Definition (Series) +1 X Let a0; a1;::: be any sequence. Then, the sum ai is called a series. i=0 Definition (Partial sum) +1 X Let n 2 N. Then, the n-th partial sum of the series ai, denoted i=0 n X Sn, is the sum ai. i=0 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 10 / 48 Sequences and series Series and partial sums Definitions Definition (Series) +1 X Let a0; a1;::: be any sequence. Then, the sum ai is called a series. i=0 Definition (Partial sum) +1 X Let n 2 N. Then, the n-th partial sum of the series ai, denoted i=0 n X Sn, is the sum ai. i=0 The partial sums themselves also form a sequence! Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 10 / 48 Theorem (Closed form of the arithmetic progression partial sum) n(a1+an) If a is an arithmetic progression, Sn = 2 . Theorem (Closed form of the geometric progression partial sum) n a1(c −1) If a is a geometric progression and c 6= 1, Sn = c−1 . Both of those theorems can be proven via (weak) mathematical induction! Sequences and series Series and partial sums Statements to prove! To kickstart the discussion on induction, here are two theorems concerning partial sums: Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 11 / 48 Both of those theorems can be proven via (weak) mathematical induction! Sequences and series Series and partial sums Statements to prove! To kickstart the discussion on induction, here are two theorems concerning partial sums: Theorem (Closed form of the arithmetic progression partial sum) n(a1+an) If a is an arithmetic progression, Sn = 2 . Theorem (Closed form of the geometric progression partial sum) n a1(c −1) If a is a geometric progression and c 6= 1, Sn = c−1 . Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 11 / 48 Sequences and series Series and partial sums Statements to prove! To kickstart the discussion on induction, here are two theorems concerning partial sums: Theorem (Closed form of the arithmetic progression partial sum) n(a1+an) If a is an arithmetic progression, Sn = 2 . Theorem (Closed form of the geometric progression partial sum) n a1(c −1) If a is a geometric progression and c 6= 1, Sn = c−1 . Both of those theorems can be proven via (weak) mathematical induction! Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 11 / 48 Weak Induction Weak Induction Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 12 / 48 Weak Induction Intro to Induction Intro to Induction Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 13 / 48 Weak Induction Intro to Induction Proof methods: The story so far... S Existential Stmt. ? - + + - Universal Existential Proof Proof Non- Indirect Constructive constructive Direct Contradiction ? ? ? Generic Particular Universal Statement Contraposition Division into cases Exhaustion Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 14 / 48 Weak Induction Intro to Induction Where induction fits S Existential Stmt. Universal Stmt - + + - Universal Existential Proof Proof Non- Indirect Constructive constructive Direct Contradiction Generic Contraposition Exhaustion Cases Particular Induction Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 15 / 48 Weak Induction Intro to Induction The penny proposition: Statement Suppose I have at least 4 in my wallet. Then, it turns out that allg my money can be stacked as 2 and 5 coins! g g Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 16 / 48 What do you think of this proof? Weak Induction Intro to Induction The penny proposition: Direct (non-inductive) proof The penny proposition Every dollar amount greater than 3 s can be paid with only 2 and 5 coins. g g g Proof (Direct, by cases). Suppose we have a total amount of C cents in our wallet. If C is an even number, then the statement is trivial by the definition of even numbers: We can just stack k 2 coins for some positive integer k. If C is an odd number greater than 3,g then it is 5 or greater. If it is 5, the problem is trivial: We only need one 5 coin. But every odd dollar amount after 5 can be retrieved by addingg any number of 2 coins, because of theg definition of parity. We are therefore done ing both cases. Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 17 / 48 Weak Induction Intro to Induction The penny proposition: Direct (non-inductive) proof The penny proposition Every dollar amount greater than 3 s can be paid with only 2 and 5 coins. g g g Proof (Direct, by cases). Suppose we have a total amount of C cents in our wallet. If C is an even number, then the statement is trivial by the definition of even numbers: We can just stack k 2 coins for some positive integer k.
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