Question

In: Advanced Math

1.Prove the following statements: . (a) If bn is recursively defined by bn =bn−1+3 for all...

1.Prove the following statements:

.

(a) If bn is recursively defined by bn =bn−1+3 for all integers n≥1 and b0 =2,

then bn =3n+2 for all n≥0.

.(b) If cn is recursively defined by cn =3cn−1+1 for all integers n≥1 and c0 =0,

then cn =(3n −1)/2 for all n≥0.

.(c) If dn is recursively defined by d0 = 1, d1 = 4 and dn = 4dn−1 −4dn−2 for all integers n ≥ 2,

then dn =(n+1)2n for all n≥0.

Solutions

Expert Solution


Related Solutions

A set M of numbers is defined recursively by 1. 2 and 3 belong to M...
A set M of numbers is defined recursively by 1. 2 and 3 belong to M 2. If x and y belong to M, so does x * y Which of the following numbers belong to M? 6, 9, 16, 21, 26, 54, 72, 218
Let {an} and {bn} be bounded sequences. Prove that limit superior {an+bn} ≦ limit superior {an}...
Let {an} and {bn} be bounded sequences. Prove that limit superior {an+bn} ≦ limit superior {an} + limit superior{bn}
Let {an} be a sequence defined recursively by a1 = 1 and an+1 = 2√ 1...
Let {an} be a sequence defined recursively by a1 = 1 and an+1 = 2√ 1 + an where n ∈ N (b) Does {an} converge or diverge? Justify your answer, making sure to cite appropriate hypotheses/theorem(s) used. Hint : Try BMCT [WHY?]. (c) (Challenge) If {an} converges then find its limit. Make sure to fully justify your answer.
Unless otherwise noted, all sets in this module are finite. Prove the following statements... 1. If...
Unless otherwise noted, all sets in this module are finite. Prove the following statements... 1. If A and B are sets then (a) |A ∪ B| = |A| + |B| − |A ∩ B| and (b) |A × B| = |A||B|. 2. If the function f : A→B is (a) injective then |A| ≤ |B|. (b) surjective then |A| ≥ |B|. 3. For each part below, there is a function f : R→R that is (a) injective and surjective. (b)...
Unless otherwise noted, all sets in this module are finite. Prove the following statements... 1. Let...
Unless otherwise noted, all sets in this module are finite. Prove the following statements... 1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r when 24|(k−r). If g : S→S is defined by (a) g(m) = f(7m) then g is injective and (b) g(m) = f(15m) then g is not injective. 2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is injective. 3....
Prove that if a language is not recursively enumerable, then its complement cannot be recursive. In...
Prove that if a language is not recursively enumerable, then its complement cannot be recursive. In this problem, you can use diagrams (black boxes with inputs and outputs to represent procedures and algorithms) as we used in class, in your proof. What can the complement be?
Prove that 3 divides n^3 −n for all n ≥ 1.
Prove that 3 divides n^3 −n for all n ≥ 1.
Prove the following statements! 1. If A and B are sets then (a) |A ∪ B|...
Prove the following statements! 1. If A and B are sets then (a) |A ∪ B| = |A| + |B| − |A ∩ B| and (b) |A × B| = |A||B|. 2. If the function f : A→B is (a) injective then |A| ≤ |B|. (b) surjective then |A| ≥ |B|. 3. For each part below, there is a function f : R→R that is (a) injective and surjective. (b) injective but not surjective. (c) surjective but not injective. (d)...
Prove the following statements! 1. Let S = {0, 1, . . . , 23} and...
Prove the following statements! 1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r when 24|(k−r). If g : S→S is defined by (a) g(m) = f(7m) then g is injective and (b) g(m) = f(15m) then g is not injective. 2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is injective. 3. Let f : A→B and g : B→C be surjective....
Prove that the function defined to be 1 on the Cantor set and 0 on the...
Prove that the function defined to be 1 on the Cantor set and 0 on the complement of the Cantor set is discontinuous at each point of the Cantor set and continuous at every point of the complement of the Cantor set.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT