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What is the generating function for the generating sequence A = 1, 9, 25, 49,…?
- a)1+(A-x2)
- b)(1-A)-1/x
- c)(1-A)+1/x2
- d)(A-x)/x3
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
What is the generating function for the generating sequence A = 1, 9, ...
The generating function for the sequence A. Using differencing:
A = 1 + 9x + 25x2 + 49x3 + ⋯(1)
−xA = 0 + x + 9x2 + 25x3 + 49x4 + ⋯(2)
(1−x)A = 1 + 8x + 16x2 + 24x3 +⋯. Since 8x + 16x2 + 24x3 + ⋯ = (1-x)A-1 ⇒ 8 + 16x + 24x2 +…= (1-A)-1/x.
A = 1 + 9x + 25x2 + 49x3 + ⋯(1)
−xA = 0 + x + 9x2 + 25x3 + 49x4 + ⋯(2)
(1−x)A = 1 + 8x + 16x2 + 24x3 +⋯. Since 8x + 16x2 + 24x3 + ⋯ = (1-x)A-1 ⇒ 8 + 16x + 24x2 +…= (1-A)-1/x.
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What is the generating function for the generating sequence A = 1, 9, ...
To find the generating function for the sequence A = 1, 9, 25, 49, ..., we can start by observing that the sequence is the square of the odd numbers. The nth term of the sequence can be expressed as (2n-1)^2.
The generating function for the sequence can be written as:
A(x) = (1x^2) + (9x^4) + (25x^6) + (49x^8) + ...
Now, let's factor out the common term x^2 from each term:
A(x) = x^2(1 + 9x^2 + 25x^4 + 49x^6 + ...)
Next, we can recognize that the series inside the parentheses is a geometric series with a common ratio of (x^2)^2 = x^4 and a first term of 1:
A(x) = x^2(1/(1 - x^4))
Therefore, the generating function for the sequence A = 1, 9, 25, 49, ... is A(x) = x^2/(1 - x^4).
The generating function for the sequence can be written as:
A(x) = (1x^2) + (9x^4) + (25x^6) + (49x^8) + ...
Now, let's factor out the common term x^2 from each term:
A(x) = x^2(1 + 9x^2 + 25x^4 + 49x^6 + ...)
Next, we can recognize that the series inside the parentheses is a geometric series with a common ratio of (x^2)^2 = x^4 and a first term of 1:
A(x) = x^2(1/(1 - x^4))
Therefore, the generating function for the sequence A = 1, 9, 25, 49, ... is A(x) = x^2/(1 - x^4).
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What is the generating function for the generating sequence A = 1, 9, 25, 49,…?a)1+(A-x2)b)(1-A)-1/xc)(1-A)+1/x2d)(A-x)/x3Correct answer is option 'B'. Can you explain this answer?
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