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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 ....
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 Winter '08
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