Homework Answers
Solve and show work for problem 8 Problem 8. Consider the sequence defined by ao =...
Solve and show work for problem 8
Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2. Problem 1. Let n 2 k. Prove that there are ktS(n, k) surjective functions (n]lk Problem 2. Let n 2 3. Find and prove an explicit formula for the Stirling numbers of the second kind S(n, n-2). Problem 3. Let n 2...
3. The sequence bois defined as follows: boo, and for integers n 2 2, bn V1...
3. The sequence bois defined as follows: boo, and for integers n 2 2, bn V1 (a) Calculate ba, ba, b4 and bs. (b) Use part (a) to guess a formula for bn for all integers n 2 0. (c) Prove by induction on n that your guess in part (b) is correct. Reflect in ePerttolio Downloard MacBook Air 80
Discrete Mathematics 3. The sequence bo, bi, b2, is defined as follows: bo 0, bnd for...
Discrete Mathematics
3. The sequence bo, bi, b2, is defined as follows: bo 0, bnd for integers n 22, bn- ehne (a) Calculate b2, b3, ba and bs (b) Use part (a) to guess a formula for bn for all integers n 20. c) Prove by induction on n that your guess in part (b) is correct.
Let ao 2 bo > 0, and consider the sequences an and bn defined by an...
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
6. Consider the recurrence relation T(n) = 2T(n-1) + 5 for integers n 1 and T(O)...
6. Consider the recurrence relation T(n) = 2T(n-1) + 5 for integers n 1 and T(O) = 0. Find a closed-form solution Using induction, prove your solution correct for all integers n 20.
6. [8 marks Consider the sequence 10, 11, ... defined by To = 0, 11 =...
6. [8 marks Consider the sequence 10, 11, ... defined by To = 0, 11 = 1, and ri = i-1 + 24-2 for all i > 2. Consider the following algorithm that attempts to compute the value In, given an n > 0. Algorithm 6 1: //pre: (n e Z) ^ (n > 0) 2: if n == 0 then 3: bro 1: else 5: a 0 6: 5+ 1 it1 //LoopInv: while i<n do ct a+b atb bc...
: Let a1, a2, a3, . . . be the sequence of integers defined by a1 = 1 and defined for n ≥ 2 by the recurrence relation a...
: Let a1, a2, a3, . . . be the sequence of integers defined by a1 = 1 and defined for n ≥ 2 by the recurrence relation an = 3an−1 + 1. Using the Principle of Mathematical Induction, prove for all integers n ≥ 1 that an = (3 n − 1) /2 .
all three questions please. thank you Prove that for all n N, O <In < 1....
all three questions please. thank you
Prove that for all n N, O <In < 1. Prove by induction that for all n EN, ER EQ. Prove that in} is convergent and find its limit l. The goal of this exercise is to prove that [0, 1] nQ is not closed. Let In} be a recursive sequence defined by In+1 = -) for n > 1, and x = 1. Prove that for all ne N, 0 <In < 1....
2. The Fibonacci numbers are defined recursively as follows: fo = 0, fi = 1 and...
2. The Fibonacci numbers are defined recursively as follows: fo = 0, fi = 1 and fn fn-l fn-2 for all n > 2. Prove that for all non-negative integers n: fnfn+2= (fn+1)2 - (-1)"
2. The Fibonacci numbers are defined recursively as follows: fo = 0, fi = 1 and fn fn-l fn-2 for all n > 2. Prove that for all non-negative integers n: fnfn+2= (fn+1)2 - (-1)"
3. (12 points) Consider the following sum: n Sn = {(i + 1)(i +2) i=0 (a)...
3. (12 points) Consider the following sum: n Sn = {(i + 1)(i +2) i=0 (a) Use properties of summations to find a closed form expression for Sn. Simplify your answer into a polynomial with rational coefficients. Show your work, and clearly indicate your final answer. (b) Use weak induction to prove that your closed form works for every integer n > 0. Make sure you include all three parts, and label them appropriately!
Solve and show work for problem 8
Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2. Problem 1. Let n 2 k. Prove that there are ktS(n, k) surjective functions (n]lk Problem 2. Let n 2 3. Find and prove an explicit formula for the Stirling numbers of the second kind S(n, n-2). Problem 3. Let n 2...
3. The sequence bois defined as follows: boo, and for integers n 2 2, bn V1 (a) Calculate ba, ba, b4 and bs. (b) Use part (a) to guess a formula for bn for all integers n 2 0. (c) Prove by induction on n that your guess in part (b) is correct. Reflect in ePerttolio Downloard MacBook Air 80
Discrete Mathematics
3. The sequence bo, bi, b2, is defined as follows: bo 0, bnd for integers n 22, bn- ehne (a) Calculate b2, b3, ba and bs (b) Use part (a) to guess a formula for bn for all integers n 20. c) Prove by induction on n that your guess in part (b) is correct.
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
6. Consider the recurrence relation T(n) = 2T(n-1) + 5 for integers n 1 and T(O) = 0. Find a closed-form solution Using induction, prove your solution correct for all integers n 20.
6. [8 marks Consider the sequence 10, 11, ... defined by To = 0, 11 = 1, and ri = i-1 + 24-2 for all i > 2. Consider the following algorithm that attempts to compute the value In, given an n > 0. Algorithm 6 1: //pre: (n e Z) ^ (n > 0) 2: if n == 0 then 3: bro 1: else 5: a 0 6: 5+ 1 it1 //LoopInv: while i<n do ct a+b atb bc...
all three questions please. thank you
Prove that for all n N, O <In < 1. Prove by induction that for all n EN, ER EQ. Prove that in} is convergent and find its limit l. The goal of this exercise is to prove that [0, 1] nQ is not closed. Let In} be a recursive sequence defined by In+1 = -) for n > 1, and x = 1. Prove that for all ne N, 0 <In < 1....
2. The Fibonacci numbers are defined recursively as follows: fo = 0, fi = 1 and fn fn-l fn-2 for all n > 2. Prove that for all non-negative integers n: fnfn+2= (fn+1)2 - (-1)"
2. The Fibonacci numbers are defined recursively as follows: fo = 0, fi = 1 and fn fn-l fn-2 for all n > 2. Prove that for all non-negative integers n: fnfn+2= (fn+1)2 - (-1)"
3. (12 points) Consider the following sum: n Sn = {(i + 1)(i +2) i=0 (a) Use properties of summations to find a closed form expression for Sn. Simplify your answer into a polynomial with rational coefficients. Show your work, and clearly indicate your final answer. (b) Use weak induction to prove that your closed form works for every integer n > 0. Make sure you include all three parts, and label them appropriately!
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