 # 1. Prove that 3n &lt; n! if n is an integer greater than 6. 2....

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1. Prove that 3n < n! if n is an integer greater than 6. 2. Use mathematically induction to prove that 9 divides n3 + (n+1)3 + (n+2)3 whenever n is a nonnegative integer. 3. Let P(n) be the statement that a postage of n cents can be formed using just 4-cent stamps and 7-cents stamps. We want to prove that P(n) is true for n ≥ 18 using strong induction proof. a) Show statements P(18), P(19), P(20) and P(21) are true, completing the basis step of the proof. b) What is the inductive hypothesis? c) What do you need to prove in the inductive step? d) Complete the inductive step for k ≥ 21. 4. Evaluate the sum ∑ = + n k k k 1 ( 1)! for n = 1, 2, 3, 4 and 5. Make a conjecture about a formula for this sum for general n, and prove your conjecture by mathematical induction. 5. Prove by induction that : nx x for all real numbers x n 1 + ≤ (1+ ) , > -1 and integers n ≥ 2.

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