HW3 - • n-1 n-1 2 n-1 =(2 n-2 n-1 • If n ≥ 2 then 2...

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Name: Math 100 Due Wednesday, October 24, 2:00 pm Homework 3 1. (2 points) Prove that, if n N , then 10 n 1(mod 3) and 10 n 1(mod 9). 2. (2 points) For every n N , let a n = 3 n 2 . Define: s n = n X i =0 a n . Find rational numbers a,b,c such that: s n = n 3 + an 2 + bn + c (Hint: find s 0 , s 1 , s 2 , and solve a system of linear equations). Prove, using induction, that your formula is correct. 3. (2 points) Prove that, if n N and n 2, then 2 ( n 2 ) > n n . Here are some hints to guide you along the way:
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Unformatted text preview: • ( n-1) ( n-1) · 2 ( n-1) = (2 n-2) n-1 . • If n ≥ 2, then 2 n-2 ≥ n . • If n ≥ 1, then 2 n ≥ n . 4. (2 points) Define a sequence d ,d 1 ,d 2 ,... recursively as follows: • Let d = 1, d 1 = 1, and d 2 = 1. • If n ≥ 3, then let d n = d n-1 + d n-2 + d n-3 . Prove that if n ∈ N , and n ≥ 1, then d n < 3 n ....
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