Functions Generation - Discrete Mathematics Questions and Answers

This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Discrete Probability – Generating Functions”.

1. What is the sequence depicted by the generating series 4 + 15x² + 10x³ + 25x⁵ + 16x⁶+⋯?
a) 10, 4, 0, 16, 25, …
b) 0, 4, 15, 10, 16, 25,…
c) 4, 0, 15, 10, 25, 16,…
d) 4, 10, 15, 25,…
View Answer

Answer: c
Explanation: Consider the coefficients of each xⁿ term. So a₀=4, since the coefficient of x₀ is 4 (x₀=1 so this is the constant term). Since 15 is the coefficient of x², so 15 is the term a₂ of the sequence. To find a₁ check the coefficient of x₁ which in this case is 0. So a₁=0. Continuing with these we have a₂=15, a₃=10, a₄=25, and a₅=16. So we have the sequence 4, 0, 15, 10, 25, 16,…

2. What is the generating function for the sequence 1, 6, 16, 216,….?
a) \(\frac{(1+6x)}{x^3}\)
b) \(\frac{1}{(1-6x)}\)
c) \(\frac{1}{(1-4x)}\)
d) 1-6x²
View Answer

Answer: b
Explanation: For the sequence 1, 6, 36, 216,… the generating function must be \(\frac{1}{(1-6x}\), when basic generating function: \(\frac{1}{1-x}\).

3. What is the generating function for generating series 1, 2, 3, 4, 5,… ?
a) \(\frac{2}{(1-3x)}\)
b) \(\frac{1}{(1+x)}\)
c) \(\frac{1}{(1−x)^2}\)
d) \(\frac{1}{(1-x2)}\)
View Answer

Answer: c
Explanation: Basic generating function is \(\frac{1}{1-x}\). If we differentiate term by term in the power series, we get (1 + x + x² + x³ +⋯)′ = 1 + 2x + 3x² + 4x³ +⋯ which is the generating series for 1, 2, 3, 4,….

4. 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
View Answer

Answer: b
Explanation: 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.

5. What is the recurrence relation for the sequence 1, 3, 7, 15, 31, 63,…?
a) a_n = 3a_n-1−2a_n+2
b) a_n = 3a_n-1−2a_n-2
c) a_n = 3a_n-1−2a_n-1
d) a_n = 3a_n-1−2a_n-3
View Answer

Answer: b
Explanation: The recurrence relation for the sequence 1, 3, 7, 15, 31, 63,… should be a_n = 3a_n-1−2a_n-2. The solution for A: A=1/1 − 3x + 2x².

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6. What is multiplication of the sequence 1, 2, 3, 4,… by the sequence 1, 3, 5, 7, 11,….?
a) 1, 5, 14, 30,…
b) 2, 8, 16, 35,…
c) 1, 4, 7, 9, 13,…
d) 4, 8, 9, 14, 28,…
View Answer

Answer: a
Explanation: The first constant term is 1⋅1, next term will be 1⋅3 + 2⋅1 = 5, the next term: 1⋅5 + 2⋅3 + 3⋅1 = 14, another one: 1⋅7 + 2⋅5 + 3⋅3 + 4⋅1 = 30. The resulting sequence is 1, 5, 14, 30,…

7. What will be the sequence generated by the generating function 4x/(1-x)²?
a) 12, 16, 20, 24,…
b) 1, 3, 5, 7, 9,…
c) 0, 4, 8, 12, 16, 20,…
d) 0, 1, 1, 3, 5, 8, 13,…
View Answer

Answer: c
Explanation: The sequence should be 0, 4, 8, 12, 16, 20,…for the generating function 4x/(1-x)², when basic generating function: 1/(1-x).

8. What is the generating function for the sequence with closed formula a_n=4(7ⁿ)+6(−2)ⁿ?
a) (4/1−7x)+6!
b) (3/1−8x)
c) (4/1−7x)+(6/1+2x)
d) (6/1-2x)+8
View Answer

Answer: c
Explanation: For the given sequence after evaluating the formula the generating formula will be (4/1−7x)+(6/1+2x).

9. Suppose G is the generating function for the sequence 4, 7, 10, 13, 16, 19,…, the find a generating function (in terms of G) for the sequence of differences between terms.
a) (1−x)G−4/x
b) (1−x)G−4/x³
c) (1−x)G+6/x
d) (1−x)G−x²
View Answer

Answer: a
Explanation: (1−x)G = 4 + 3x + 6x² + 9x³ +⋯ which can be accepted. We can compute it like this:
3 + 6x + 9x² + ⋯ = (1−x)G−4/x.

10. Find the sequence generated by 1/1−x²−x⁴.,assume that 1, 1, 2, 3, 5, 8,… has generating function 1/1−x−x².
a) 0, 0, 1, 1, 2, 3, 5, 8,…
b) 0, 1, 2, 3, 5, 8,…
c) 1, 1, 2, 2, 4, 6, 8,…
d) 1, 4, 3, 5, 7,…
View Answer

Answer: a
Explanation: Based on the given generating function, the sequence will be 0, 0, 1, 1, 2, 3, 5, 8,… which is generated by 1/1−x²−x⁴.

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