HW3 - ( n-1) ( n-1) 2 ( n-1) = (2 n-2) n-1 . If n 2, then 2...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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:
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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) Dene 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 ....
View Full Document

Ask a homework question - tutors are online