Unit 5
Patterns with a rule, and the sums they generate. The distinction between an explicit and a recursive rule runs through the whole unit.
Lesson 5.1
An arithmetic sequence adds a fixed difference from one term to the next. A geometric sequence multiplies by a fixed ratio. For a nonzero geometric starting term, ratios between successive terms reveal the pattern; differences reveal arithmetic patterns. Sequence inputs are discrete integer positions.
Lesson 5.2
An explicit rule gives a term directly from its position. A recursive rule gives an initial term and a method for obtaining later terms from previous ones. A recursive rule without a starting value generally does not specify a unique sequence.
Lesson 5.3
A series adds sequence terms. Finite arithmetic and geometric sums have different formulas. An infinite geometric series has a finite sum only when the common ratio has absolute value less than 1; in that case its terms approach zero fast enough for the partial sums to settle.
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