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Quiz

MAT102 quiz

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School
University of Toronto Mississauga
Department
Mathematics
Course
MAT102H5
Professor
Shay Fuchs
Semester
Fall

Description
MAT102S - Intro. to Mathematical Proofs - Winter 2011 - UTM Quiz 3 (Version B) - SOLUTIONS 3 1. [4 marks] Prove by induction that for all n 2 N, n ▯ 4n is divisible by 3. Solution: 3 3 Basic step) For n = 1, n ▯ 4n = 1 ▯ 4 = ▯3 and ▯3 is divisible by 3. Induction step) Suppose that n ▯4n is divisible by 3. We need to show that then (n+1) ▯4(n+1) is also divisible by 3. In order to do so, we rearrange the expression (n + 1) ▯ 4(n + 1) as (n + 1) ▯ 4(n + 1) = n + 3n + 3n + 1 ▯ 4n ▯ 4 = n ▯ 4n + (3n + 3n ▯ 3): By induction assumption we know that n ▯ 4n is divisible by 3. The term (3n + 3n ▯ 3) is also divisible by 3. Therefore (n + 1) ▯ 4(n + 1) is divisible by 3. 2. [4 marks] Prove by induction that for all n 2 N, 2 2 2 2 1 1 + 2 + 3 + ▯▯▯ + n = 6n(1 + n)(1 + 2n): Solution: 2 1 Basic step) For n = 1, 1 = 1 and 6(1)(1 + 1)(1 + 2) = 1 so the formula is TRUE. Induction step) Suppose that the formula is true for n. Then we have 2 2 2 2 2 1 2 1 + 2 + 3 + ▯▯▯ + n + (n + 1) = 6n(1 + n)(1 + 2n) + (n + 1) 2 n(1 + n)(1 + 2n) + 6(n + 1) = 6
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