To prove: That r = 2 x and s = − x 2 for second order homogeneous recurrence relation a n = r a n − 1 + s a n − 2 , n ≥ 2 , with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root x .

Question

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Answer 1

Question

Chapter 5.3, Problem 26E

(a)

To determine

To prove: That r=2x and s=−x2 for second order homogeneous recurrence relation an=ran−1+san−2,n≥2, with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root x.

(b)

To determine

To prove: That if c is any constant then an=cnxn is a solution of the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

(c)

To determine

To prove: Theorem 5.3.1 using (a) and (b), for the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

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Answer 2

Question

Chapter 5.3, Problem 26E

(a)

To determine

To prove: That r=2x and s=−x2 for second order homogeneous recurrence relation an=ran−1+san−2,n≥2, with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root x.

(b)

To determine

To prove: That if c is any constant then an=cnxn is a solution of the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

(c)

To determine

To prove: Theorem 5.3.1 using (a) and (b), for the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

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Answer 3

Question

Chapter 5.3, Problem 26E

(a)

To determine

To prove: That r=2x and s=−x2 for second order homogeneous recurrence relation an=ran−1+san−2,n≥2, with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root x.

(b)

To determine

To prove: That if c is any constant then an=cnxn is a solution of the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

(c)

To determine

To prove: Theorem 5.3.1 using (a) and (b), for the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

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Answer 4

Question

Chapter 5.3, Problem 26E

(a)

To determine

To prove: That r=2x and s=−x2 for second order homogeneous recurrence relation an=ran−1+san−2,n≥2, with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root x.

(b)

To determine

To prove: That if c is any constant then an=cnxn is a solution of the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

(c)

To determine

To prove: Theorem 5.3.1 using (a) and (b), for the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

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Answer 5

Chapter 5.3, Problem 26E

(a)

To determine

To prove: That r=2x and s=−x2 for second order homogeneous recurrence relation an=ran−1+san−2,n≥2, with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root x.

(b)

To determine

To prove: That if c is any constant then an=cnxn is a solution of the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

(c)

To determine

To prove: Theorem 5.3.1 using (a) and (b), for the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

Answer 6

Chapter 5.3, Problem 26E

(a)

To determine

To prove: That r=2x and s=−x2 for second order homogeneous recurrence relation an=ran−1+san−2,n≥2, with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root x.

Answer 7

Chapter 5.3, Problem 26E

(a)

To determine

To prove: That r=2x and s=−x2 for second order homogeneous recurrence relation an=ran−1+san−2,n≥2, with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root x.

Answer 8

(b)

To determine

To prove: That if c is any constant then an=cnxn is a solution of the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

Answer 9

(b)

To determine

To prove: That if c is any constant then an=cnxn is a solution of the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

Answer 10

(c)

To determine

To prove: Theorem 5.3.1 using (a) and (b), for the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

Answer 11

(c)

To determine

To prove: Theorem 5.3.1 using (a) and (b), for the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

Answer 12

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Answer 13

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Answer 14

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Answer 15

Ch. 5.1 - True/False Questions The statement i=1n(2i1)=n2...Ch. 5.1 - Prob. 2TFQ Ch. 5.1 - Prob. 3TFQ Ch. 5.1 - Prob. 4TFQ Ch. 5.1 - Prob. 5TFQ Ch. 5.1 - Prob. 6TFQ Ch. 5.1 - Prob. 7TFQ Ch. 5.1 - Prob. 8TFQ Ch. 5.1 - Prob. 9TFQ Ch. 5.1 - Prob. 10TFQ

To prove: That r = 2 x and s = − x 2 for second order homogeneous recurrence relation a n = r a n − 1 + s a n − 2 , n ≥ 2 , with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root x .

Solution Summary: The author demonstrates that r=2x and s=-x2 for second order homogeneous recurrence relation.

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To prove: That r=2x and s=−x2 for second order homogeneous recurrence relation an=ran−1+san−2,n≥2, with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root x.

To prove: That if c is any constant then an=cnxn is a solution of the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

To prove: Theorem 5.3.1 using (a) and (b), for the recurrence relation an=ran−1+san−2,n≥2 with constant coefficients. Assume that its characteristic polynomial has just one (repeated) nonzero root.

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Chapter 5 Solutions