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11.7 – Proof by Mathematical
Induction
• Mathematical induction is a method of
proving statements involving natural numbers.
• Mathematical induction is a method of
proving statements involving natural numbers.
• To prove a statement true for all natural
numbers, n:
• Mathematical induction is a method of
proving statements involving natural numbers.
• To prove a statement true for all natural
numbers, n:
1. Show that the statement is true for n = 1
• Mathematical induction is a method of
proving statements involving natural numbers.
• To prove a statement true for all natural
numbers, n:
1. Show that the statement is true for n = 1
2. Assume that the statement is true for
some natural number, k. (induction hypothesis)
• Mathematical induction is a method of
proving statements involving natural numbers.
• To prove a statement true for all natural
numbers, n:
1. Show that the statement is true for n = 1
2. Assume that the statement is true for
some natural number, k. (induction hypothesis)
3. Show that the statement is true for the
next natural number, k + 1.
Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2
4
Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2
4
1. Show that the statement is true for n = 1.
Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2
4
1. Show that the statement is true for n = 1.
13 + 23 + 33 + … + n3 = n2(n+1)2
4
Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2
4
1. Show that the statement is true for n = 1.
13 + 23 + 33 + … + n3 = n2(n+1)2
4
n = 1:
Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2
4
1. Show that the statement is true for n = 1.
13 + 23 + 33 + … + n3 = n2(n+1)2
4
n = 1:
13= 12(1+1)2
4
Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2
4
1. Show that the statement is true for n = 1.
13 + 23 + 33 + … + n3 = n2(n+1)2
4
n = 1:
13= 12(1+1)2
4
Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2
4
1. Show that the statement is true for n = 1.
13 + 23 + 33 + … + n3 = n2(n+1)2
4
n = 1:
13= 12(1+1)2
4
1 = 1(2)2
4
Ex. 1 Prove that 13 + 23 + 33 + … + n3 = n2(n+1)2
4
1. Show that the statement is true for n = 1.
13 + 23 + 33 + … + n3 = n2(n+1)2
4
n = 1:
13= 12(1+1)2
4
1 = 1(2)2
4
1 = 1, thus shown to be true.
2. Assume that the statement is true for some
natural number, k.
2. Assume that the statement is true for some
natural number, k.
13 + 23 + 33 + … + n3 = n2(n+1)2
4
2. Assume that the statement is true for some
natural number, k.
13 + 23 + 33 + … + n3 = n2(n+1)2
4
13 + 23 + 33 + … + k3 = k2(k+1)2
4
3. Show that the statement is true for the next
natural number, k + 1.
3. Show that the statement is true for the next
natural number, k + 1.
13 + 23 + 33 + … + k3 = k2(k+1)2
4
3. Show that the statement is true for the next
natural number, k + 1.
13 + 23 + 33 + … + k3 = k2(k+1)2
4
13 + 23 + 33 + … + k3 + (k+1)3 = k2(k+1)2 + (k+1)3
4
3. Show that the statement is true for the next
natural number, k + 1.
13 + 23 + 33 + … + k3 = k2(k+1)2
4
13 + 23 + 33 + … + k3 + (k+1)3 = k2(k+1)2 + (k+1)3
4
= k2(k+1)2 + (k+1)3
4
1
3. Show that the statement is true for the next
natural number, k + 1.
13 + 23 + 33 + … + k3 = k2(k+1)2
4
13 + 23 + 33 + … + k3 + (k+1)3 = k2(k+1)2 + (k+1)3
4
= k2(k+1)2 + (k+1)3 . 4
4
1
4
3. Show that the statement is true for the next
natural number, k + 1.
13 + 23 + 33 + … + k3 = k2(k+1)2
4
13 + 23 + 33 + … + k3 + (k+1)3 = k2(k+1)2 + (k+1)3
4
= k2(k+1)2 + (k+1)3 . 4
4
1
4
= k2(k+1)2 + 4(k+1)3
4
4
= k2(k+1)2 + 4(k+1)3
4
= k2(k+1)2 + 4(k+1)3
4
= (k+1)2 [k2 + 4(k+1)]
4
= k2(k+1)2 + 4(k+1)3
4
= (k+1)2 [k2 + 4(k+1)]
4
= (k+1)2 [k2 + 4k+4]
4
= k2(k+1)2 + 4(k+1)3
4
= (k+1)2 [k2 + 4(k+1)]
4
= (k+1)2 [k2 + 4k+4]
4
= (k+1)2 (k+2)2
4
= k2(k+1)2 + 4(k+1)3
4
= (k+1)2 [k2 + 4(k+1)]
4
= (k+1)2 [k2 + 4k+4]
4
= (k+1)2 (k+2)2
4
13 + 23 + 33 + … + k3 + (k+1)3 = (k+1)2 (k+2)2
4
• So we started with the problem:
13 + 23 + 33 + … + n3 = n2(n+1)2
4
• So we started with the problem:
13 + 23 + 33 + … + n3 = n2(n+1)2
4
• And we showed:
13 + 23 + 33 + … + k3 + (k+1)3 = (k+1)2 (k+2)2
4
• So we started with the problem:
13 + 23 + 33 + … + n3 = n2(n+1)2
4
• And we showed:
13 + 23 + 33 + … + k3 + (k+1)3 = (k+1)2 (k+2)2
4
• So we proved that the statement holds true
when we replace n with k + 1 and thus proven.
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
n = 1:
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
n = 1: 12 = 1(3∙1 – 1)
2
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
n = 1: 12 = 1(3∙1 – 1)
2
1=1
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
n = 1: 12 = 1(3∙1 – 1)
2
1=1
n = 2:
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
n = 1: 12 = 1(3∙1 – 1)
2
1=1
n = 2: 12 + 22 = 2(3∙2 – 1)
2
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
n = 1: 12 = 1(3∙1 – 1)
2
1=1
n = 2: 12 + 22 = 2(3∙2 – 1)
2
5=5
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
n = 1: 12 = 1(3∙1 – 1)
2
1=1
n = 2: 12 + 22 = 2(3∙2 – 1)
2
5=5
n = 3:
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
n = 1: 12 = 1(3∙1 – 1)
2
1=1
n = 2: 12 + 22 = 2(3∙2 – 1)
2
5=5
n = 3: 12 + 22 + 32 = 3(3∙3 – 1)
2
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
n = 1: 12 = 1(3∙1 – 1)
2
1=1
n = 2: 12 + 22 = 2(3∙2 – 1)
2
5=5
n = 3: 12 + 22 + 32 = 3(3∙3 – 1)
2
14 = 12
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
n = 1: 12 = 1(3∙1 – 1)
2
1=1
n = 2: 12 + 22 = 2(3∙2 – 1)
2
5=5
n = 3: 12 + 22 + 32 = 3(3∙3 – 1)
2
14 = 12, FALSE
Ex. 2 Find a counterexample to disprove the statement.
12 + 22 + 32 + … + n2 = n(3n – 1)
2
n = 1: 12 = 1(3∙1 – 1)
2
1=1
n = 2: 12 + 22 = 2(3∙2 – 1)
2
5=5
n = 3: 12 + 22 + 32 = 3(3∙3 – 1)
2
14 = 12, FALSE
n=3