Warm-up Problem• Prove by induction: Every n≥8 is the sum of 3’s and 5’s “n is triquinqual”... Prove m+1 is triquinqual... Prove m+1 is triquinqual.
Trang 1Warm-up Problem
• Prove by induction: Every n≥8 is the sum of 3’s and 5’s
Trang 2Warm-up Problem
• Prove by induction: Every n≥8 is the sum of 3’s and 5’s (“n is triquinqual”)
Trang 3Warm-up Problem
• Prove by induction: Every n≥8 is the sum of 3’s and 5’s (“n is triquinqual”)
• Proof:
o n =
Trang 4Warm-up Problem
• Prove by induction: Every n≥8 is the sum of 3’s and 5’s (“n is triquinqual”)
• Proof:
o n = 8 =
Trang 5Warm-up Problem
• Prove by induction: Every n≥8 is the sum of 3’s and 5’s (“n is triquinqual”)
• Proof:
o n = 8 = 3+5 is triquinqual
o n = 9
Trang 6Warm-up Problem
• Prove by induction: Every n≥8 is the sum of 3’s and 5’s (“n is triquinqual”)
• Proof: Need multiple base cases
o n = 8 = 3+5 is triquinqual
o n = 9
Trang 7Warm-up Problem
• Prove by induction: Every n≥8 is the sum of 3’s and 5’s (“n is triquinqual”)
• Proof: Need multiple base cases
o n = 8 = 3+5 is triquinqual
o n = 9 = 3+3+3 is triquinqual
o n =
Trang 8Warm-up Problem
• Prove by induction: Every n≥8 is the sum of 3’s and 5’s (“n is triquinqual”)
• Proof: Need multiple base cases
o n = 8 = 3+5 is triquinqual
o n = 9 = 3+3+3 is triquinqual
o n = 10 = 5+5 is triquinqual
o (I.H.) Assume
Trang 9Warm-up Problem
• Prove by induction: Every n≥8 is the sum of 3’s and 5’s (“n is triquinqual”)
• Proof: Need multiple base cases
o n = 8 = 3+5 is triquinqual
o n = 9 = 3+3+3 is triquinqual
o n = 10 = 5+5 is triquinqual
o (I.H.) Assume m ≥ 10 and every n≤m is triquinqual Prove
Trang 10Warm-up Problem
• Prove by induction: Every n≥8 is the sum of 3’s and 5’s (“n is triquinqual”)
• Proof: Need multiple base cases
o n = 8 = 3+5 is triquinqual
o n = 9 = 3+3+3 is triquinqual
o n = 10 = 5+5 is triquinqual
o (I.H.) Assume m ≥ 10 and every n≤m is triquinqual Prove m+1 is triquinqual.
Trang 11Warm-up Problem
• Prove by induction: Every n≥8 is the sum of 3’s and 5’s (“n is triquinqual”)
• Proof: Need multiple base cases
o n = 8 = 3+5 is triquinqual
o n = 9 = 3+3+3 is triquinqual
o n = 10 = 5+5 is triquinqual
o (I.H.) Assume m ≥ 10 and every n≤m is triquinqual Prove m+1 is triquinqual.
o m+1 is triquinqual because m+1 = (m-2)+3 and m-2 ≥ 8 is triquinqual.