Item: tExplicitGeometric3a

Stem:

The first four terms of a geometric sequence are given below. Choose the algebraic expression that represents the nth term in this sequence.

Key:

• Observable: 1. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's incorrect. Consider the case where n = 1. Then the expression you selected evaluates to 1/2(3n-1) = 1/2(31-1) = 1/2. But the first term is equal to 3, not 1/2, so this cannot be the correct expression to describe the sequence.
  • Teacher:The student may have misremembered the formula for finding the nth term of a geometric sequence. The student should be encouraged to verify expressions by substituting numbers from the problem.
• Observable: 2. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's incorrect. Consider the case where n = 1. Then the expression you selected evaluates to 1/2n + 3 = 1/21 + 3 = 3.5. But the first term is equal to 3, not 3.5, so this cannot be the correct expression to describe the sequence.
  • Teacher:The student may have misremembered the formula for finding the nth term of a geometric sequence. The student should be encouraged to verify expressions by substituting numbers from the problem.
• Observable: 3. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's incorrect. Consider the case where n = 1. Then the expression you selected evaluates to n/2 = 1/2. But the first term is equal to 3, not 1/2, so this cannot be the correct expression to describe the sequence.
  • Teacher:The student can probably identify the common ratio of a geometric sequence, but does not know how to represent this in an explicit formula for the nth term.
• Observable: 4. Score: 1; Evidence Value: 1.
  • Student:Right! You have identified the correct expression for the nth term.
  • Teacher:The student successfully identified the correct explicit formula for an increasing geometric sequence that includes fractions in the terms.
• Observable: 5. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's incorrect. Consider the case where n = 1. Then the expression you selected evaluates to n/2 + 3 = 3.5. But the first term is equal to 3, not 3.5, so this cannot be the correct expression to describe the sequence.
  • Teacher:The student has not identified the correct expression for the nth term of a geometric sequence, and should be encouraged to verify expressions by substituting numbers from the problem.

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