1. a) Deduce from the relation Fk+1F-1-F2 = (-1)k, that for any positive integer k > 2 Fk b) Deduce from part a) that for any positive integer n, Fn+1 = Σ 1)k+1 1. a) Deduce from the rela...
The Fibonacci Sequence F1, F2, ... of
integers is defined recursively by F1=F2=1
and Fn=Fn-1+Fn-2 for each integer
. Prove
that (picture) Just the top one( not
7.23)
n 3 Chapter 7 Reviewing Proof Techniques 196 an-2 for every integer and an ao, a1, a2,... is a sequence of rational numbers such that ao = n > 2, then for every positive integer n, an- 3F nif n is even 2Fn+1 an = 2 Fn+ 1 if n is odd....
Let Fn be a free group of rank n. (a) Show that Fn contains a subgroup isomorphic to Fk whenever 1 <k <n. (b) Show that F2 contains subgroups isomorphic to Fk for all k > 2, and hence that Fn contains subgroups isomorphic to Fk for all k > 1. (c) Can an infinite group be generated by two elements of finite order? If so, then give an example. If not, then explain why not.
3. The sequence (Fn) of Fibonacci numbers is defined by the recursive relation Fn+2 Fn+1+ F for all n E N and with Fi = F2= 1. to find a recursive relation for the sequence of ratios (a) Use the recursive relation for (F) Fn+ Fn an Hint: Divide by Fn+1 N (b) Show by induction that an 1 for all n (c) Given that the limit l = lim,0 an exists (so you do not need to prove that...
The Fibonacci numbers are defined as follows, f1=1, f2=1 and
fn+2=fn+fn+1 whenever n>= 1.
(a) Characterize the set of integers n for which fn is even and
prove your answer using induction
(b) Please do b as well.
The Fibonacci numbers are defined as follows: fi -1, f21, and fn+2 nfn+1 whenever n 21. (a) Characterize the set of integers n for which fn is even and prove your answer using induction. (b) Use induction to prove that Σ. 1...
DEFINITION: For a positive integer n, τ(n) is the number of
positive divisors of n and σ(n) is the sum of those divisors.
4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...
2. Find a formula for Σ LVk] when n is a positive integer.
2. Find a formula for Σ LVk] when n is a positive integer.
1, and let σ be a permutation of {1, , n). Recall that for each integer m a) Let n 1, we denote ơm--σ ο . . . o σ. Show that n times b) Let 21, and let be a permutation of..,n consisting of a unique cycle of length n. Deduce from the previous question that there exists i e (1,..., n) such that i +c() )+22(n1).
1, and let σ be a permutation of {1, , n). Recall...
3. Consider the recurrence relation an = 80n/2 + n², where n=2", for some integer k. a) Give a big-O estimate for an. b) What is the recurrence relation for the sequence bk obtained from an by doing the substitution n= n=2k ?
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...
(a) Let n be any positive integer. Briefly explain (no formal proofs) why n > 1 ≡ ¬(n = 1). (b) Recall that a positive integer p is prime iff there do not exist a positive integers n and m, both greater than 1, such that p = nm. (I.e., Prime(p) means ¬∃n ∃m (n > 1 ∧ m > 1 ∧ p = nm).) Give a formal proof of the following: for any prime p, any positive integers n...