Explanation Given: nis a positive integer. Proof: Leta n represent the sum of the firstnsquares: a n = a n − 1 + n 2 . Let G ( x ) = ∑ k = 0 + ∞ a k x k = a 0 + ∑ k = 1 + ∞ a k x k = a 0 + ∑ k = 1 + ∞ ( a k − 1 + k 2 ) x k = 0 + ∑ k = 1 + ∞ a k − 1 x k − 1 + ∑ k = 1 + ∞ k 2 x k = x ∑ k = 0 + ∞ To determine (b) To prove: Use part (a) to find an explicit formula for the sum 1 2 + 2 2 +· · ·+ n 2 .

8th Edition

ISBN: 9781307447118

Author: ROSEN

Publisher: MCG

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Chapter 8.4, Problem 46E

To determine

(a)

To prove:

( x² +x )/(1 -x )4 is the generating function for the sequence { a_n }, wherea_n = 1² + 2² +· · ·+ n².

To determine

(b)

To prove:

Use part (a) to find an explicit formula for the sum 1² + 2² +· · ·+ n².

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Chapter 8.4, Problem 45E

Chapter 8.4, Problem 47E

Chapter 8 Solutions

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(a) To prove: ( x 2 +x )/(1 -x )4 is the generating function for the sequence { a n }, wherea n = 1 2 + 2 2 +· · ·+ n 2 .

Solution Summary: The author explains that (x 2 +x)/(1 -x )4 is the generating function for the sequence a n.

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(a)

To prove:

( x² +x )/(1 -x )4 is the generating function for the sequence { a_n }, wherea_n = 1² + 2² +· · ·+ n².

(b)

To prove:

Use part (a) to find an explicit formula for the sum 1² + 2² +· · ·+ n².

Want to see the full answer?

Chapter 8 Solutions

(a) To prove: ( x 2 +x )/(1 -x )4 is the generating function for the sequence { a n }, wherea n = 1 2 + 2 2 +· · ·+ n 2 .

Solution Summary: The author explains that (x 2 +x)/(1 -x )4 is the generating function for the sequence a n.

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(a) To prove: ( x2 +x )/(1 -x )4 is the generating function for the sequence { an }, wherean = 12 + 22 +· · ·+ n2.

(b) To prove: Use part (a) to find an explicit formula for the sum 12 + 22 +· · ·+ n2.

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Chapter 8 Solutions

(a)

To prove:

( x² +x )/(1 -x )4 is the generating function for the sequence { a_n }, wherea_n = 1² + 2² +· · ·+ n².

(b)

To prove:

Use part (a) to find an explicit formula for the sum 1² + 2² +· · ·+ n².