0 Yes, There Are Proofs! 1 Logic 2 Sets And Relations 3 Functions 4 The Integers 5 Induction And Recursion 6 Principles Of Counting 7 Permutations And Combinations 8 Algorithms 9 Graphs 10 Paths And Circuits 11 Applications Of Paths And Circuits 12 Trees 13 Planar Graphs And Colorings 14 The Max Flow - Min Cut Theorem expand_more
5.1 Mathematical Induction 5.2 Recursively Defined Sequences 5.3 Solving Recurrence Relations; The Characteristic Polynomial 5.4 Solving Recurrence Relations; Generating Functions Chapter Questions expand_more
Problem 1RE: Use mathematical induction to show that i=1ni(i!)=(n+1)!1. Problem 2RE: Using mathematical induction, show that
for all .
Problem 3RE: Using mathematical induction, show that (112)n1n2 for all n1. Problem 4RE: Prove that for all integers.
Problem 5RE: 5. Use mathematical induction to prove that is divisible by 3 for all.
Problem 6RE: 6. Prove that for all.
Problem 7RE Problem 8RE: 8. (a) Give an example of a function with domaina subset of with the property that for all .
(b)... Problem 9RE: Give a recursive definition of each of the following sequences: 1,5,29,173,1037,........... Problem 10RE: Guess a simple formula for each of the following products and prove that your guess is correct for... Problem 11RE: 11. Consider the sequence defined by and for. What is ?
Problem 12RE: 12. Find the sum.
Problem 13RE: 13. Let be defined recursively by and, for , . Prove that for all integers.
Problem 14RE: Define f:ZZ by f(a)=34a, and for tZ define a sequence a1,a2,a3,........ by a1=f(t) and, for k1,... Problem 15RE: Consider the arithmetic sequence that begins 5,9,13. Find the 32nd and 100th terms of this sequence.... Problem 16RE: 16. The first two terms of a sequence are 6 and 2.
(a) If the sequence is arithmetic, find the 27th... Problem 17RE: 17. Let be the first four terms of an arithmetic sequence. Show that and use this fact to factor the... Problem 18RE: Explain why the sum of 500 terms of the series (32)742(32)740+(32)738........ 913[(32)742(23)258]. Problem 19RE: 19. (a) Define the Fibonacci sequence.
(b) Is it possible for three successive terms in the... Problem 20RE: Show that, for n2, the nth term of the Fibonacci sequence is less than (74)n1. [use the definition... Problem 21RE: Let f1,f2,....... be the Fibonacci sequence as defined in 5.2.3. Prove that... Problem 22RE: Suppose you walk up a flight of stairs one or two steps at a time. In how many ways can you reach... Problem 23RE: 23. Solve the recurrence relation given that and Use the characteristic polynomial as described in... Problem 24RE: Solve Exercise 23 using the method of generating functions described in Section 5.4. Problem 25RE: 25. Find a formula for, given and for .
Problem 26RE: Let an be the sequence defined by a0=2,a1=1, and an=6an19an2 for n2. Find the first five terms of... Problem 27RE Problem 28RE Problem 29RE Problem 30RE: 30. (For students of calculus) Let denote the Fibonacci sequence as defined in 5.2.3. Evaluate... format_list_bulleted