1. Chapter 1: Number Patterns 1.6: Geometric Sequences
Essential Question: What is a geometric sequence?

2. 1.6: Geometric Sequences {3, 9, 27, 81, …}
Back in section 1.4, we talked about arithmetic sequences. An arithmetic sequence was a sequence that simply added a constant term, d.
Geometric sequences (a.k.a. geometric progression) are sequences where a common ratio, r, is multiplied to successive terms.
Examples:
{3, 9, 27, 81, …}
r = 3
r = ½

3. 1.6: Geometric Sequences Recursive Form:
Recursive form for arithmetic sequence:
un = un-1 + d, for n ≥ 2
RECURSIVE FORM FOR GEOMETRIC SEQUENCE:
un = run-1, for r≠0 and n ≥ 2
Remember, two things are necessary for a recursive function
Starting point (u1) and the function (un)

4. 1.6: Geometric Sequences Explicit Form
If there is a constant number being multiplied over and over, it’s the same as multiplying that common ratio as an exponent
Ex: u2 = u1 ∙ r u3 = u2 ∙ r = (u1 ∙ r) ∙ r = u1 ∙ r2 u4 = u3 ∙ r = (u1 ∙ r2) ∙ r = u1 ∙ r3
This gives us the explicit form: un = u1 ∙ rn-1

5. 1.6: Geometric Sequences Example 4: Explicit
Write the explicit form of a geometric sequence where the first two terms are 2 and -2/5 and find the first five terms of the sequence.
First, we need to find the common ratio, acquired by dividing successive terms:
Explicit Form: un = 2 ∙ (-1/5)n-1
The sequence begins

6. 1.6: Geometric Sequences Example 5: Explicit Form
The 4th term and 9th terms of a geometric sequence are 20 and Find the explicit form.
The first thing we need to do is figure out the common ratio.
The 4th term: u4 = u1rn-1
20=u1r3
The 9th term: u9 = u1rn =u1r8
Their ratio can be used to find r:

7. 1.6: Geometric Sequences Example 5 (Continued) u4=20, u9=-640
We now know r=-2
Find u1 by using u4:
un = u1(-2)n-1
u4 =u1(-2)3
20 =u1(-8)
-5/2=u1
We have everything we need for our sequence: un = -5/2 ∙ (-2)n-1

8. 1.6: Geometric Sequences Partial Sums
The kth partial sum of the geometric sequence {un} with common ratio ≠ 1 is
We can also calculate the partial sums using the sum feature of the calculator:
sum seq(function,x,1,k) where we plug our function in

9. 1.6: Geometric Sequences Example 6: Partial Sums Find the sum:
The first term is -3/2, and the common ratio is ½
This is the 9th partial sum (9 terms) of the geometric sequence:
Which you can plug into the formula from the last page, or use the calculator:
sum seq((-3/2)*(-1/2)^(x-1),x,1,9) =

10. 1.6: Geometric Sequences Exercises Page 63-64 1 – 17 23 – 33 39 – 41
odd problems