Question

. Consider the sequence defined recursively as a0 = 5, a1 = 16 and ak =...

. Consider the sequence defined recursively as a0 = 5, a1 = 16 and ak = 7ak−1 − 10ak−2 for all integers k ≥ 2. Prove that an = 3 · 2 n + 2 · 5 n for each integer n ≥ 0

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

We are going to prove by strong induction, that the given recursive sequence satisfies .

For n=0, and for n=1, .

Now suppose, the given recursive sequence satisfies the answer for all , that is, for all for some . Then, .

Hence, we have proved by induction that the given recursive sequence satisfies .

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Suppose that a sequence an (n = 0,1,2,...) is defined recursively by a0 = 1, a1...
Suppose that a sequence an (n = 0,1,2,...) is defined recursively by a0 = 1, a1 = 7, an = 4an−1 − 4an−2 (n ≥ 2). Prove by induction that an = (5n + 2)2n−1 for all n ≥ 0.
2. Exercise 19 section 5.4. Suppose that a1, a2, a3, …. Is a sequence defined as...
2. Exercise 19 section 5.4. Suppose that a1, a2, a3, …. Is a sequence defined as follows: a1=1 ak=2a⌊k/2⌋ for every integer k>=2. Prove that an <= n for each integer n >=1. plzz send with all the step
Consider the sequence defined recursively by an+1 = (an + 1)/2 if an is an odd...
Consider the sequence defined recursively by an+1 = (an + 1)/2 if an is an odd number an+1 = an/2 if an is an even number (a) Let a0 be equal to the last digit in your student number, and compute a1, a2, a3, a4. (b) Suppose an = 1, and find an+4. (c) If a0 = 4, does limn→∞ an exist?
Consider a sequence defined recursively as X0= 1,X1= 3, and Xn=Xn-1+ 3Xn-2 for n ≥ 2....
Consider a sequence defined recursively as X0= 1,X1= 3, and Xn=Xn-1+ 3Xn-2 for n ≥ 2. Prove that Xn=O(2.4^n) and Xn = Ω(2.3^n). Hint:First, prove by induction that 1/2*(2.3^n) ≤ Xn ≤ 2.8^n for all n ≥ 0 Find claim, base case and inductive step. Please show step and explain all work and details
1.13. Let a1, a2, . . . , ak be integers with gcd(a1, a2, . ....
1.13. Let a1, a2, . . . , ak be integers with gcd(a1, a2, . . . , ak) = 1, i.e., the largest positive integer dividing all of a1, . . . , ak is 1. Prove that the equation a1u1 + a2u2 + · · · + akuk = 1 has a solution in integers u1, u2, . . . , uk. (Hint. Repeatedly apply the extended Euclidean algorithm, Theorem 1.11. You may find it easier to prove...
4. Let an be the sequence defined by a0 = 0 and an = 2an−1 +...
4. Let an be the sequence defined by a0 = 0 and an = 2an−1 + 2 for n > 1. (a) Find the value of sum 4 i=0 ai . (b) Use induction to prove that an = 2n+1 − 2 for all n ∈ N.
10. Let P(k) be the following statement: ”Let a1, a2, . . . , ak be...
10. Let P(k) be the following statement: ”Let a1, a2, . . . , ak be integers and p be a prime. If p|(a1 · a2 · a3 · · · ak), then p|ai for some i with 1 ≤ i ≤ k.” Prove that P(k) holds for all positive integers k
1) Suppose a1, a2, a3, ... is a sequence of integers such that a1 =1/16 and...
1) Suppose a1, a2, a3, ... is a sequence of integers such that a1 =1/16 and an = 4an−1. Guess a formula for an and prove that your guess is correct. 2) Show that given 5 integer numbers, you can always find two of the numbers whose difference will be a multiple of 4. 3) Four cats and five mice form a row. In how many ways can they form the row if the mice are always together? Please help...
Let (a_n)∞n=1 be a sequence defined recursively by a1 = 1, a_n+1 = sqrt(3a_n) for n...
Let (a_n)∞n=1 be a sequence defined recursively by a1 = 1, a_n+1 = sqrt(3a_n) for n ≥ 1. we know that the sequence converges. Find its limit. Hint: You may make use of the property that lim n→∞ b_n = lim n→∞ b_n if a sequence (b_n)∞n=1 converges to a positive real number.  
Given this pseudocode:   input: sequence of numbers ak k, length of sequence answer := a1 for...
Given this pseudocode:   input: sequence of numbers ak k, length of sequence answer := a1 for i = 2 to k if (ai > answer), then answer = ai End-for What is the value of answer for the sequence {-1, 4, -7, 10, 2} with k = 5? Please provide detailed answer! Thank you!
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT