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Lesson Synthesis
Choose one arithmetic sequence and one geometric sequence that students encountered in this lesson, and write the closed-form de nition for each next to its recursive form de nition. Ask students to discuss with a partner how the two forms are alike and di erent.
If you created a semi-permanent classroom display a few lessons ago showing examples of arithmetic and geometric sequences with their recursive de nitions, add the closed-form de nitions for the same sequences to the display.
7.4 Close Up
Cool Down: 5 minutes
Student Task Statement
A sequence is de ned by . Write a closed-form de nition for this sequence.
Student Response
(or equivalent)
Student Lesson Summary
A closed form de nition of a sequence is a formula that can directly compute any term of the sequence.
Here’s an arithmetic sequence: 6, 10, 14, 18, 22, . . . In this sequence, each term is 4 more than the previous term. Another way to write this sequence is
Each term has an increasing number of fours. Since you know , then for any
Geometric sequences behave the same way, but with repeated multiplication. The geometric sequence can be written as . Then its closed form de nition is
Closed form de nitions are used to directly compute terms that would be hard to nd with a recursive de nition.
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Teacher Guide Algebra