Item: tRecursiveRuleGeometric3a

Stem:

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

6, -18, 54, -162, . . .

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

Options:

Key:

• Observable: 1. Score: 1; Evidence Value: 1.
  • Student:Right! Good job. Starting with the second term, each term in the sequence is equal to negative three times the previous term.
  • Teacher:The student was able to select the correct recursive rule for generating successive terms in a geometric sequence with integer terms and a negative common ratio.
• Observable: 2. 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 = 6, and -3 n-1 = -30 = 1. So an+1 = 6 * 1 = 6.But an+1 = a2 = -18. 6 is not equal to -18, 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 geometric sequence with integer terms and a negative common ratio. The student may have confused the recursive rule for a geometric sequence with an explicit rule for a geometric sequence.
• 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 = 6. So an+1 = -3 + 6 = 3.But an+1 = a2 = -18. 3 is not equal to -18, 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 geometric sequence with integer terms and a negative common ratio. The student may be confusing multiplicative and additive operations, or confusing the rules for geometric sequences with those for arithemetic sequences.
• 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 = 6. So an+1 = 6/6 = 1. But an+1 = a2 = -18. 1 is not equal to -18, 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 geometric sequence with integer terms and a negative 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 = 6. So an+1 = 6 * 6 = 36. But an+1 = a2 = -18. 36 is not equal to -18, 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 geometric sequence with integer terms and a negative common ratio. The student may have confused the initial term with the common ratio.

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