Item: tRecursiveRuleGeometric2b

Stem:

Let a1, a2, a3, ..., an, ... be a sequence of integers. The first four terms of the sequence are given below.

-3, -6, -12, -24, . . .

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

Options:

Key:

• Observable: 2. Score: 1; Evidence Value: 1.
  • Student:Right! Good job. Starting with the second term, each term in the sequence is equal to two times the previous term.
  • Teacher:The student was able to select the correct recursive rule for generating successive terms in a decreasing geometric sequence with negative integer terms.
• Observable: 1. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's incorrect. Try testing the equation you selected by substituting a few numbers. Suppose n = 1. Then an = -3. So an+1 = -2 * -3 = 6. But an+1 = a2 = -6. 6 is not equal to -6, so this cannot be the correct equation.
  • Teacher:The student was not able to select the correct recursive rule for generating successive terms in a decreasing geometric sequence with negative integer terms. The student may have difficulty with multiplicative operations involving negative numbers.
• Observable: 3. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's incorrect. Try testing the equation you selected by substituting a few numbers. Suppose n = 1. Then an = -3, and 2n-1 = 20 = 1. So an+1 = -3 * 1 = -3. But an+1 = a2 = -6. -3 is not equal to -6, so this cannot be the correct equation.
  • Teacher:The student was not able to select the correct recursive rule for generating successive terms in a decreasing geometric sequence with negative integer terms. The student may have confused the recursive rule for a geometric sequence with an explicit rule for a geometric sequence.
• Observable: 4. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's incorrect. Try testing the equation you selected by substituting a few numbers. Suppose n = 1. Then an = -3. So an+1 = -3 * -3 = 9. But an+1 = a2 = -6. 9 is not equal to -6, so this cannot be the correct equation.
  • Teacher:The student was not able to select the correct recursive rule for generating successive terms in a decreasing geometric sequence with negative integer terms. The student may have confused the initial term with the common ratio.
• Observable: 5. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's incorrect. Try testing the equation you selected by substituting a few numbers. Suppose n = 1. Then an = -3. So an+1 = -(-3) / 3 = 1. But an+1 = a2 = -6. 1 is not equal to -6, so this cannot be the correct equation.
  • Teacher:The student was not able to select the correct recursive rule for generating successive terms in a decreasing geometric sequence with negative integer terms.

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