Chapter 1: Sequences and Series

3. 1.3 Geometric Sequences

Objectives:

1. Identify assumptions made when identifying a geometric sequence or series.
   
2. Provide and justify an example of a geometric sequence.
   
3. Derive a rule for determining the general term of a geometric sequence.
   
4. Determine the first term, the common ratio, the number of terms, or the value of a specific term in a problem involving a geometric sequence.
   

In a Geometric Sequence, the ratio of consecutive terms is a constant.

Examples:

• 3, 6, 12, 24, 48                   
• 200, 100, 50, 25, ...      
• 4, -8, 16, -32, 64, ...

For a geometric sequence, the common ratio is represented by r, 

and the general term is given by  t_n = t₁r^{n - 1}.

For the Examples above, the general terms are:

• t_n = 3⋅$\cdot$2^{n - 1}
• t_n = 200⋅$\cdot$(12$\frac{1}{2}$)^{n - 1}
• t_n = 4⋅$\cdot$(-2)^{n - 1}