CS1802 Optional Recitation Week 11-12: Series, Induction, Recurrences

Transcription

1 CS1802 Discrete Structures Recitation Fall 2017 Nov 19 - November 26, 2017 CS1802 Optional Recitation Week 11-12: Series, Induction, Recurrences 1 Sequences, Series and Recurrences PB 1. Prove that n k=0 ( n+k ) 1 k = 2 n 2 k 1

2 PB 2 ; part A requires combinatorics or calculus) Prove that the sequence a n = (1 + 1 n )n is monotonically increasing and that the sequence b n = (1 + 1 n )n+1 is monotonically decreasing; yet all a-s are smaller than all b-s. Specifically, prove that for any n (1 + 1 n )n < (1 + 1 n + 1 )n+1 < (1 + 1 n + 1 )n+2 < (1 + 1 n )n+1 2

3 part B) Conclude that the two sequences have the same limit, the base of natural logarithm e, and that 1 n + 1 ln(1 + 1 n ) 1 n PB 3 ; easier with convexity argument. In a 1, a 2,..., a n are positive numbers then their mean inequality states that the order from big to small is : quadratic mean, arithmetic mean, geometric mean, harmonic mean: 1 n ( k a 2 k ) 1 n ( k a k ) ( k a k ) 1/n n k 1 a k 3

4 PB 4 ; requires series integral test. Prove that the inverse-n log(n) series diverges : 1 n log(n) = k=2 PB 5 Solve asymptotically the following recurrence (say a running time for a recursive algorithm). Asymptotically means we only care about the order of growth for function T, not the exact coefficients. T (n) = 4T (n/2) + n 4

5 PB 6 Solve asymptotically the following recurrence (say a running time for a recursive algorithm). Asymptotically means we only care about the order of growth for function T, not the exact coefficients. T (n) = 4T (n/4) + n PB 7 Solve asymptotically the following recurrence (say a running time for a recursive algorithm). Asymptotically means we only care about the order of growth for function T, not the exact coefficients. T (n) = T (n/2) + T (n/4) + T (n/8) + n 5

6 2 Induction proofs PB 8 Sum of Perfect Squares. Prove that for all natural number n, 10 n can be written as a sum of two perfect squares ( 10 n = a 2 + b 2 for some a, b positive integers). PB 9 Josephine s Problem In Josephine s Kingdom every woman has to pass a logic exam before being allowed to marry. Every married woman knows about the fidelity of every man in the Kingdom except for her own husband, and etiquette demands that no woman should be told about the fidelity of her husband. Also, a gunshot fired in any house in the Kingdom will be heard in any other house. Queen Josephine announced that at least one unfaithful man had been discovered in the Kingdom, and that any woman knowing her husband to be unfaithful was required to shoot him at midnight following the day after she discovered his infidelity. How did the wives manage this? 6

7 PB 10 2n dots are placed around the outside of the circle; n of them are colored red and the remaining n are colored blue. Going around the circle anticlockwise, you keep a count of how many red and blue dots you have passed. If at all times the number of red dots you have passed is at least the number of blue dots, you consider it a successful trip around the circle. Prove that no matter how the dots are placed on the circle, it is possible to have a successful trip around the circle if you start at the right point. PB 11. Lines dividing a plane If n lines are drawn on a plane, and no two lines are parallel, and no 3 lines are concurrent, show that they dive plane into (n 2 + n + 2)/2 regions. 7

8 PB 12 FF If n lines are drawn on a plane, and no two lines are parallel, and no 3 lines are concurrent, they dive plane into (n2 + n + 2)/2 regions. Show that it is possible to color the regions formed with only two colors so that no two adjacent regions (a common side) share the same color. PB 13 Towers of Hanoi lower bound, F. The Towers of Hanoi task consists on three towers, towers B and C empty, and tower A containing the disks of radius 1 to n on top of each other (see picture). BIGGER DISKS ARE NOT ALLOWED ON TOP OF SMALLER DISKS AT ANY TIME. The task is to move all disks to tower B, one disk at a time, allowing any moves between towers. Prove that the minimum number of moves required to complete the task is 2n 1. 8

9 PB 14 A sphere is covered with some number of caps which are hemispheres. Prove that it is possible to choose four hemispheres, and remove all others, while still keeping the sphere covered. PB 15. Show that for any 3D convex polyhedra we have #FACES + #VERTICES = #EDGES + 2 9

10 PB 16. Prove the Inclusion-Exclusion Principle for n sets, by induction over n PB 17 Fermat s Theorem. Prove Fermat s little theorem, by induction over a (use binomial theorem) For any p prime and reminder 0 < a < p, we have a p 1 = 1 mod p 10