Math 1 · Unit 9
Arithmetic and Geometric Sequences
Five lessons: a sequence is a function, arithmetic sequences (lines made of dots), geometric sequences (exponentials made of dots), recursive and explicit rules both ways, and building sequences from the real world. Adding gives arithmetic and multiplying gives geometric: the two families you already know, written as lists.
A Sequence Is a Function
A sequence is a list in order, and it is a function: the term number goes in and the term comes out. Only whole-number inputs, so the graph is separate dots. The notation a sub n and f(n) say the same thing. Adding gives arithmetic, multiplying gives geometric, and some lists are neither.
Start Unit 9, Lesson 1Arithmetic Sequences, Lines Made of Dots
Adding gives arithmetic. The common difference d, the recursive rule (the stairs) and the explicit rule (the elevator), and why the explicit rule uses n minus 1. The common difference is the slope, the rule simplifies to slope-intercept form, and the y-intercept is the zeroth term, not the first.
Start Unit 9, Lesson 2Geometric Sequences, Exponentials Made of Dots
Multiplying gives geometric. The common ratio r, found by dividing; growth when r is above 1 and decay when it is between 0 and 1. Recursive and explicit rules, with n minus 1 because the first term has not been multiplied yet. The common ratio is the growth factor from Unit 7, and the terms are dots on the curve.
Start Unit 9, Lesson 3Recursive and Explicit, Both Ways
Two rules for one list. Recursive is the stairs: Next = Now + 12, starting at 4, or in formal notation a sub n plus 1 = a sub n + 12 with a sub 1 = 4. Explicit is the elevator. Change either into the other in three steps, for arithmetic and geometric, and see why both are worth knowing.
Start Unit 9, Lesson 4Building Sequences from the Real World
Story, table, or picture to a rule to an answer. Decide arithmetic or geometric, write the rule, and use it: pizza toppings, an orchard, stacked cups, splitting bacteria, and the interest balances from Unit 8. Always ask what n counts and whether the counting starts at 0 or at 1.
Start Unit 9, Lesson 5Printables for Unit 9
Study guides, worksheets with answer keys, and lesson plans for every lesson in this unit.