Quadratic Sequences and Second Differences

Mathematics · Algebra · ages 15-16

Name ______________________ Date ____________

The first differences of a sequence are 2, 4, 6, 8, 10. Its second difference is 2.
Circle one: True False
In a quadratic sequence, the second difference is always twice the n squared coefficient.
Circle one: True False
The nth term of a sequence is 3n squared + 2n + 5. What is its second difference?
Answer: ______________
A quadratic sequence has a constant second difference of 8. What is the coefficient of n squared in its rule?
- 2
- 8
- 4
- 16
A sequence with a constant first difference of 4 is quadratic.
Circle one: True False
The rule for a sequence is 2n squared + 3n + 1. What is the fifth term?
Answer: ______________
Find the nth term of the sequence 7, 13, 21, 31, 43.
- n squared + 5n + 1
- n squared + 3n + 3
- n squared + 3n + 5
- n squared + 4n + 2
The rule for a sequence is n squared + 2n + 3. What is the fifth term?
- 27
- 38
- 33
- 45
A quadratic sequence starts 4, 11, ... and has a constant second difference of 6. What is its fifth term?
- 68
- 58
- 74
- 46
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

Answer key

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

True · Each first difference grows by 2, so the second difference, the difference between them, is 2 throughout.
True · That is the core link: a rule with n squared has second difference 2, with 2n squared it is 4, and so on.
6 · The second difference is twice the n squared coefficient, and the coefficient is 3, so it is 6.
4 · The second difference is twice the n squared coefficient, so halve 8 to get 4.
False · A constant first difference is the fingerprint of a linear sequence, not a quadratic one. A quadratic sequence has a constant second difference instead.
66 · Substitute n = 5: 2 times 25 is 50, plus 15, plus 1, which is 66.
n squared + 3n + 3 · The second difference is 2, so a is 1. Peel off the n squared terms to leave 6, 9, 12, 15, 18, a linear sequence with rule 3n + 3. The full rule is n squared + 3n + 3.
38 · Substitute n = 5: 25 + 10 + 3 = 38.
68 · Halve the second difference to get a = 3. The first differences are 7, 13, 19, 25, so the terms are 4, 11, 24, 43, 68.
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.