Assignment_5

# Assignment_5 - n> 4n for all integers n ≥ 0 6 Let b 1...

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Sequences & Induction Due: September 29 1. Let n be a positive number. Give an expression using summation notation for the sum of the cubes of the first n even integers. Then show that this sum is equal to 2n 2 (n+1) 2 for all positive n. 2. Consider the following sequence: 2, 12, 22, 32,... Give an expression using summation notation for the sum of the first n terms in this sequence. Then use induction to prove that this sum is equal to 5n 2 - 3n for all positive numbers n. 3. Consider the following sequence: 0,3,8,15,24,35,48,... Give an explicit formula for computing the k th number in this sequence. (Hint: it only depends on k.) Then use induction to prove that the sum of the first n numbers in this sequence is equal to (2n 3 + 3n 2 + n - 6)/6 for all positive numbers n. 4. Prove by mathematical induction that 2 n < (n+2)! for all n 0. 5. Let a 0 =6 and a k =a k-1 +4 for all k>0. Prove by induction that a
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Unformatted text preview: n > 4n for all integers n ≥ 0. 6. Let b 1 =3, b 2 =5 and b k =3b k-1- 2b k-2 for all k>2. Prove by induction that b n = 2 n +1 for all integers n ≥ 1. 7. Suppose we have N deck chairs on board the Titanic and it ʼ s our job to stack them all together into one pile. We can take any pile (of one or more chairs) and place it on top of another pile to create a new pile. Assuming none of the chairs is initially stacked on any other, how many steps (a step is the act of creating one new pile out of two) does it take us to stack all N chairs? Give a proof using strong induction. Then go hop in a lifeboat. 8. Prove using induction that the following statement holds for all n ≥ 0: if n is odd then 9 n mod 10 = 9 and if n is even then 9 n mod 10 = 1. COMP SCI 360 Assignment #5 Penn State University Fall 2010 PAGE 1 OF 1...
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