9.3 Geometric Sequences and Series!

CLARIFICATION

Sequence- commas

2, 4, 8, 16...

Series- plus signs/ addition

2 + 4 + 8 + 16...

Geometric sequence- ratios of consecutive terms are the same

Example:       5, 15, 45, 135, 405...           r = 3

The number r is the common ratio.  It cannot be zero, otherwise, it is not a geometric sequence. When r is negative, the signs will alternate.  This means that if the first term is positive, the next will be negative, then positive, then negative and so on.  

(The common difference of an arithmetic sequence is like the common ratio of a geometric sequence)

Recursive Formula

Explicit Formula (exponential)

Example: find the nth term 

Sum of Finite Geometric Sequences

Sum of an Infinite Geometric Series

We can only find the sum of an infinite geometric series if .  If the common ratio was greater than, then the sum would be infinity.  This is because adding an infinite amount of continuously increasing numbers together will be infinity.  

This is the graph of a geometric sequence with a common ratio less than one (2/3).  The red line represents the numbers in the sequence.  The blue line represents the sum of the numbers.  See how the blue line begins to curve then levels off?  This is because the numbers keep getting smaller and smaller.