Chapter 1: Number Patterns1.6: Geometric SequencesEssential Question: What is a geometric sequence?

1.6: Geometric Sequences• 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, …}▫

5 5 5 5, , , ,...2 4 8 16

r = 3

r = ½

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)

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

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

25 1

2 5r

2 2 2 22, , , , ,...

5 25 125 625

1.6: Geometric Sequences•Example 5: Explicit Form

▫The 4th term and 9th terms of a geometric sequence are 20 and -640. 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-1

-640=u1r8

Their ratio can be used to find r:

813

1

5

640

20

32

2

u r

u r

r

r

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

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

11

1

1

kk

nn

ru u

r

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) =

3 3 3 3 3 3 3 3 3

2 4 8 16 32 64 128 256 512

13 1

2 2

n

1.6: Geometric Sequences

•Exercises▫Page 63-64

1 – 17 23 – 33 39 – 41 odd problems