Final - SOLUTIONS FOR THE FINAL 1 (8 points) : Prove that,...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: SOLUTIONS FOR THE FINAL 1 (8 points) : Prove that, for any positive integer n , 7 divides 11 n 4 n . Define Proposition P n : 7 divides 11 n 4 n . We show by induction that P n is true for every n . The base of induction is easy to verify: when n = 1, 7 divides 11 1 4 1 = 7. The inductive step consists of showing that, if 7 divides 11 n 4 n , then 7 also divides 11 n +1 4 n +1 . Observe that 11 n +1 4 n +1 = 11 11 n 4 4 n = (7 + 4)11 n 4 4 n = 7 11 n + 4(11 n 4 n ) . In the right hand side, 7 11 n is clearly a multiple of 7, and 4(11 n 4 n ) is a multiple of 7 by the induction hypothesis. Thus, the left hand side must be divisible by 7. 2 (12 points) : Compute the following limits: (a) (6 points) lim n + n 4 / 2 n 2 n 1 = 1 2 . We have lim n + n 4 / 2 n 2 n 1 = lim n (1 + n 3 / 2 n ) n (2 1 /n ) = lim 1 + n 3 / 2 n 2 1 /n = 1 + lim n 3 / 2 n 2 lim1 /n We know that lim n 3 / 2 n = 0 (we proved in class that lim n p /a n = 0 for any a > 1), and lim1 /n = 0. Therefore, lim n + n 4 / 2 n 2 n 1 = 1 2 + 0 = 1 2 . (b) (6 points) lim ( radicalbig n 2 + 6 n radicalbig n 2 + 2 n ) = 2 . Note that radicalbig n 2 + 6 n radicalbig n 2 + 2 n = ( radicalbig n 2 + 6 n radicalbig n 2 + 2 n ) n 2 + 6 n + n 2 + 2 n n 2 + 6 n + n 2 + 2 n = n 2 + 6 n 2 n 2 + 2 n 2 n radicalbig 1 + 6 /n + n radicalbig 1 + 2 /n = 4 n n ( radicalbig 1 + 6 /n + radicalbig 1 + 2 /n ) = 4 radicalbig 1 + 6 /n + radicalbig 1 + 2 /n ....
View Full Document

Page1 / 3

Final - SOLUTIONS FOR THE FINAL 1 (8 points) : Prove that,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online