Item: tRecursiveRuleGeometric3b

Stem:

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

-2, 10, -50, 250, . . .

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

Options:

Key:

• Observable: 5. Score: 1; Evidence Value: 1.
  • Student:Right! Good job. Starting with the second term, each term in the sequence is equal to negative five 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: 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 = -2. So an+1 = -2/2 = -1. But an+1 = a2 = 10. -1 is not equal to 10, 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: 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 = -2, and -5 n-1 = -50 = 1. So an+1 = -2 * 1 = -2. But an+1 = a2 = 10. -2 is not equal to 10, 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 = -2. So an+1 = -5 + -2 = -7. But an+1 = a2 = 10. -7 is not equal to 10, 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 = -2. So an+1 = -2 * -2 = 4. But an+1 = a2 = 10. 4 is not equal to 10, 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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