Item: tRecursiveRuleOtherRecursive2b

Stem:

Let a1, a2, a3,... an,... be a sequence of positive numbers. The first five terms of the sequence are given below.

9, 12, 18, 27, 39, ...

Select the correct rule for finding the (n+1)th term from the nth term.

Options:

1. an+1 = nan

2. an+1 = an + 3

3. an+1 = an + 9

4. an+1 = an + 3n

5. an+1 = an + 12

Key:

• Observable: 1. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's not right. Think about the following example: The first term is 9, so when n = 1, an = 9. 1 * 9 = 9. But the next term in the sequence is 12, not 9. So this rule does not apply to the sequence given in the question. It's a good approach to test your rule by checking it with several pairs of numbers from the sequence.
  • Teacher:Student answered incorrectly.
• Observable: 2. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's not right. For an = 9, the expression you selected evaluates as: 9 + 3 = 12. The next term is 12, so the rule works for these particular numbers. But it does not work in general. For example, when an = 12, 12 + 3 = 15. But the next term in the sequence is 18, not 15, so the rule fails in this case. It's a good approach to test your rule by checking it with several pairs of numbers from the sequence.
  • Teacher:Student answered incorrectly.
• Observable: 3. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's not right. For an = 18, the expression you selected evaluates as: 18 + 9 = 27. The next term is 27, so the rule works for these particular numbers. But it does not work in general. For example, when an = 12, 12 + 9 = 21. But the next term in the sequence is 18, not 21, so the rule fails in this case. It's a good approach to test your rule by checking it with several pairs of numbers from the sequence.
  • Teacher:Student answered incorrectly.
• Observable: 4. Score: 1; Evidence Value: 1.
  • Student:The expression you selected is correct. Great job!
  • Teacher:The student answered correctly.
• Observable: 5. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's not right. For an = 27, the expression you selected evaluates as: 27 + 12 = 39. The next term is 39, so the rule works for these particular numbers. But it does not work in general. For example, when an = 12, 12 + 12 = 24. But the next term in the sequence is 18, not 24, so the rule fails in this case. It's a good approach to test your rule by checking it with several pairs of numbers from the sequence.
  • Teacher:Student answered incorrectly.

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