Generating Functions MCQ

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• Find the sequence generated by 1/1−x2−x4.,assume that 1, 1, 2, 3, 5, 8,… has generating function 1/1−x−x2.
  ক

  0, 0, 1, 1, 2, 3, 5, 8,…

  খ

  0, 1, 2, 3, 5, 8,…

  গ

  1, 1, 2, 2, 4, 6, 8,…

  ঘ

  1, 4, 3, 5, 7,…
• 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.
  ক

  (1−x)G−4/x

  খ

  (1−x)G−4/x3

  গ

  (1−x)G+6/x

  ঘ

  (1−x)G−x2
• What is the generating function for the sequence with closed formula an=4(7n)+6(−2)n?
  ক

  (4/1−7x)+6!

  খ

  (3/1−8x)

  গ

  (4/1−7x)+(6/1+2x)

  ঘ

  (6/1-2x)+8
• What will be the sequence generated by the generating function 4x/(1-x)2?
  ক

  12, 16, 20, 24,…

  খ

  1, 3, 5, 7, 9,…

  গ

  0, 4, 8, 12, 16, 20,…

  ঘ

  0, 1, 1, 3, 5, 8, 13,…
• What is multiplication of the sequence 1, 2, 3, 4,… by the sequence 1, 3, 5, 7, 11,….?
  ক

  1, 5, 14, 30,…

  খ

  2, 8, 16, 35,…

  গ

  1, 4, 7, 9, 13,…

  ঘ

  4, 8, 9, 14, 28,…
• What is the recurrence relation for the sequence 1, 3, 7, 15, 31, 63,…?
  ক

  an = 3an-1−2an+2

  খ

  an = 3an-1−2an-2

  গ

  an = 3an-1−2an-1

  ঘ

  an = 3an-1−2an-3
• What is the generating function for the generating sequence A = 1, 9, 25, 49,…?
  ক

  1+(A-x2)

  খ

  (1-A)-1/x

  গ

  (1-A)+1/x2

  ঘ

  (A-x)/x3
• What is the sequence depicted by the generating series 4 + 15x2 + 10x3 + 25x5 + 16x6+⋯?
  ক

  10, 4, 0, 16, 25, …

  খ

  0, 4, 15, 10, 16, 25,…

  গ

  4, 0, 15, 10, 25, 16,…

  ঘ

  4, 10, 15, 25,…