Generating Functions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Packing a whole sequence into one function

Mathematics · Discrete Mathematics · ages 20-21
Name ______________________   Date ____________

  1. The polynomial (1 plus x) cubed is the generating function of the sequence 1, 3, 3, 1. What is the coefficient of x squared?

    • 3
    • 2
    • 6
    • 1

  2. The sequence 1, 1, 1, ... has generating function 1/(1 minus x). What is the coefficient of x to the power 5? Give a number.

    Answer: ______________

  3. The sequence 1, 1, 1, and on has generating function 1 divided by (1 minus x). What is the coefficient of x to the power 5?

    • 5
    • 1
    • 0

  4. A generating function stores a sequence as the coefficients of a power series.

    Circle one:   True   False

  5. The sequence 1, 2, 3, 4, and on has generating function x divided by (1 minus x) squared. What is the coefficient of x to the power 4?

    Answer: ______________

  6. Zoe says the product of two generating functions just adds the two sequences term by term. Is Zoe right?

    Circle one:   True   False

  7. A sequence obeys a(n) = 2 times a(n minus 1) with a0 = 1. Its generating function is 1/(1 minus 2 x). What is a4? Give a number.

    Answer: ______________

  8. A sequence obeys a(n) equal to 2 times a(n minus 1) with a0 equal 1. What is a4?

    Answer: ______________

  9. Zoe says the product of two generating functions just adds the two sequences term by term.

    Circle one:   True   False

  10. A sequence obeys b(n) equal to 3 times b(n minus 1) with b0 equal 2. What is b2?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Packing a whole sequence into one function W1-mt_WAX3O-HjRk-s1

  1. 3 · Expanding (1 plus x) cubed gives 1 plus 3 x plus 3 x squared plus x cubed, so the x squared coefficient is 3.
  2. 1 · The series for 1/(1 minus x) is 1 plus x plus x squared and so on, so every coefficient is 1.
  3. 1 · Every coefficient in that expansion equals 1.
  4. True · That storage trick is the whole definition.
  5. 4 · The expansion runs x plus 2x squared plus 3x cubed plus 4x to the 4th.
  6. False · Zoe is wrong. Multiplying generating functions convolves the sequences: each new coefficient sums products of pairs, which counts combined choices.
  7. 16 · The recurrence doubles each term from 1, giving 1, 2, 4, 8, 16, and the generating function equation confirms the coefficient of x to the power 4 is 16.
  8. 16 · Doubling from 1 gives 1, 2, 4, 8, 16.
  9. False · Products convolve: each coefficient sums products of pairs.
  10. 18 · The generating function is 2 over (1 minus 3x), expanding as 2 plus 6x plus 18x squared.
Worksheet · LightMySky