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Generating Functions

This quiz is designed to assess your understanding of generating functions, a powerful tool used to solve various problems in combinatorics and probability.

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is the generating function for the sequence (1, 2, 3, 4, \dots)?

  1. \(\frac{x}{1-x}\)
  2. \(\frac{x}{(1-x)^2}\)
  3. \(\frac{x^2}{1-x}\)
  4. \(\frac{x^2}{(1-x)^2}\)
Question 2 Multiple Choice (Single Answer)

Find the generating function for the sequence (1, 1, 2, 3, 5, 8, \dots), where each term is the sum of the previous two terms.

  1. \(\frac{x}{1-x-x^2}\)
  2. \(\frac{x}{1-2x+x^2}\)
  3. \(\frac{x}{1-x+x^2}\)
  4. \(\frac{x}{1+x+x^2}\)
Question 3 Multiple Choice (Single Answer)

What is the generating function for the sequence (1, 2, 4, 8, 16, \dots)?

  1. \(\frac{x}{1-2x}\)
  2. \(\frac{x}{1-x^2}\)
  3. \(\frac{x}{1-3x}\)
  4. \(\frac{x}{1-4x}\)
Question 4 Multiple Choice (Single Answer)

Find the generating function for the sequence (1, 3, 6, 10, 15, \dots), where each term is the sum of the first (n) positive integers.

  1. \(\frac{x}{(1-x)^3}\)
  2. \(\frac{x}{(1-x)^2}\)
  3. \(\frac{x}{(1-x)}\)
  4. \(\frac{x}{1-x+x^2}\)
Question 5 Multiple Choice (Single Answer)

What is the generating function for the sequence (1, 4, 9, 16, 25, \dots)?

  1. \(\frac{x}{1-x^2}\)
  2. \(\frac{x}{1-2x^2}\)
  3. \(\frac{x}{1-3x^2}\)
  4. \(\frac{x}{1-4x^2}\)
Question 6 Multiple Choice (Single Answer)

Find the generating function for the sequence (1, 2, 4, 7, 11, \dots), where each term is the sum of the first (n) odd positive integers.

  1. \(\frac{x}{(1-x)^4}\)
  2. \(\frac{x}{(1-x)^3}\)
  3. \(\frac{x}{(1-x)^2}\)
  4. \(\frac{x}{(1-x)}\)
Question 7 Multiple Choice (Single Answer)

What is the generating function for the sequence (1, 3, 5, 7, 9, \dots)?

  1. \(\frac{x}{1-x^3}\)
  2. \(\frac{x}{1-2x^3}\)
  3. \(\frac{x}{1-3x^3}\)
  4. \(\frac{x}{1-4x^3}\)
Question 8 Multiple Choice (Single Answer)

Find the generating function for the sequence (1, 4, 9, 16, 25, \dots), where each term is the square of the first (n) positive integers.

  1. \(\frac{x}{(1-x)^5}\)
  2. \(\frac{x}{(1-x)^4}\)
  3. \(\frac{x}{(1-x)^3}\)
  4. \(\frac{x}{(1-x)^2}\)
Question 9 Multiple Choice (Single Answer)

What is the generating function for the sequence (1, 2, 6, 24, 120, \dots)?

  1. \(\frac{x}{1-x-x^2-x^3}\)
  2. \(\frac{x}{1-2x-x^2-x^3}\)
  3. \(\frac{x}{1-3x-x^2-x^3}\)
  4. \(\frac{x}{1-4x-x^2-x^3}\)
Question 10 Multiple Choice (Single Answer)

Find the generating function for the sequence (1, 3, 6, 10, 15, \dots), where each term is the sum of the first (n) triangular numbers.

  1. \(\frac{x}{(1-x)^4}\)
  2. \(\frac{x}{(1-x)^3}\)
  3. \(\frac{x}{(1-x)^2}\)
  4. \(\frac{x}{(1-x)}\)
Question 11 Multiple Choice (Single Answer)

What is the generating function for the sequence (1, 5, 14, 30, 55, \dots)?

  1. \(\frac{x}{1-x-x^2}\)
  2. \(\frac{x}{1-2x-x^2}\)
  3. \(\frac{x}{1-3x-x^2}\)
  4. \(\frac{x}{1-4x-x^2}\)
Question 12 Multiple Choice (Single Answer)

Find the generating function for the sequence (1, 4, 10, 20, 35, \dots), where each term is the sum of the first (n) square numbers.

  1. \(\frac{x}{(1-x)^5}\)
  2. \(\frac{x}{(1-x)^4}\)
  3. \(\frac{x}{(1-x)^3}\)
  4. \(\frac{x}{(1-x)^2}\)
Question 13 Multiple Choice (Single Answer)

What is the generating function for the sequence (1, 2, 5, 12, 22, \dots)?

  1. \(\frac{x}{1-x-x^2-x^3}\)
  2. \(\frac{x}{1-2x-x^2-x^3}\)
  3. \(\frac{x}{1-3x-x^2-x^3}\)
  4. \(\frac{x}{1-4x-x^2-x^3}\)
Question 14 Multiple Choice (Single Answer)

Find the generating function for the sequence (1, 5, 15, 35, 70, \dots), where each term is the sum of the first (n) pentagonal numbers.

  1. \(\frac{x}{(1-x)^6}\)
  2. \(\frac{x}{(1-x)^5}\)
  3. \(\frac{x}{(1-x)^4}\)
  4. \(\frac{x}{(1-x)^3}\)
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