Quadratic Sequences and Second Differences · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Quadratic Sequences and Second Differences

Mathematics · Algebra · ages 15-16
Name ______________________   Date ____________

  1. What are the first differences of the sequence 6, 10, 16, 24, 34?

    • 4, 6, 8, 10
    • 2, 2, 2, 2
    • 10, 16, 24, 34
    • 4, 8, 12, 16

  2. The nth term of a sequence is 3n squared + 2n + 5. What is its second difference?

    Answer: ______________

  3. A sequence is 1, 4, 9, 16, 25. What is the second difference of this sequence?

    Answer: ______________

  4. Which row of a difference table proves that a sequence is quadratic?

    • the first difference stays constant
    • the second difference stays constant
    • the terms grow by n each time
    • the terms are all positive

  5. A quadratic sequence has a constant second difference of 6. What value should you use for a, the coefficient of n², in the rule an² + bn + c?

    Answer: ______________

  6. The rule for a sequence is n squared + 4n + 3. What is the sixth term?

    Answer: ______________

  7. Find the nth term of the sequence 5, 12, 23, 38, 57.

    • n squared + 2n + 2
    • 2n squared + 2n + 1
    • 2n squared + n + 1
    • 2n squared + n + 2

  8. Ben finds the nth term of 2, 7, 14, 23, 34. He works out the second difference is 2, so a is 1, and the remainder after peeling off the n squared terms is 1, 3, 5, 7, 9. He concludes the rule is n squared + 2n + 1. Where is the slip?

    • The second difference should be 4
    • The remainder is not a linear sequence
    • The remainder rule is 2n - 1, so the full rule is n squared + 2n - 1
    • He should have checked at n = 2 instead of n = 1

  9. Maya finds the nth term of 9, 7, 3, -3, -11. She writes the first differences as 2, 4, 6, 8, takes the second difference as 2, and concludes the rule is n squared + n + 7. Where is the slip?

    • The second difference should be 4
    • The remainder method does not work for negative terms
    • She should have started counting at n = 0
    • The first differences should be negative: -2, -4, -6, -8, so a is -1

  10. A sequence is 6, 15, 28, 45, 66. Its second difference is 4, so a = 2 in the rule an² + bn + c. Subtract 2n² from each term (2, 8, 18, 32, 50) to get a new list of numbers. What is the common difference of that new list?

    Answer: ______________

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Answer key

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

Quadratic Sequences and Second Differences W1-mt_gyZT3Tcw7U-s1

  1. 4, 6, 8, 10 · Each first difference is the jump from one term to the next: 10 - 6, 16 - 10, 24 - 16, 34 - 24.
  2. 6 · The second difference is twice the n squared coefficient, and the coefficient is 3, so it is 6.
  3. 2 · Taking differences twice gives a constant value of 2, which is the second difference.
  4. the second difference stays constant · A quadratic sequence has a constant second difference. A constant first difference would make it linear.
  5. 3 · Every step of n² adds 2 to the second difference, so a is always half the second difference.
  6. 63 · Substitute n = 6: 36 + 24 + 3 = 63.
  7. 2n squared + n + 2 · The second difference is 4, so a is 2. Peel off 2, 8, 18, 32, 50 to leave 3, 4, 5, 6, 7, whose rule is n + 2. The full rule is 2n squared + n + 2.
  8. The remainder rule is 2n - 1, so the full rule is n squared + 2n - 1 · The remainder 1, 3, 5, 7, 9 has first difference 2, so its rule is 2n - 1, not 2n + 1. Ben's rule gives 4 at n = 1, but the first term is 2.
  9. The first differences should be negative: -2, -4, -6, -8, so a is -1 · The sequence is decreasing, so the first differences are -2, -4, -6, -8. The second difference is -2, a is -1, and the true rule is -n squared + n + 9.
  10. 3 · Once the n² part is stripped out, what's left behaves like a plain arithmetic sequence, and its common difference is b.
Worksheet · LightMySky