11.3 Geometric Sequences

1
9-3 Geometric Sequences Series
2
Geometric Sequence

• The ratio of a term to its previous term is
  constant.
• This means you multiply by the same number to get
  each term.
• This number that you multiply by is called the
  common ratio (r).

3
Example Decide whether each sequence is
geometric.

• 4,-8,16,-32,
• -8/4-2
• 16/-8-2
• -32/16-2
• Geometric (common ratio is -2)

• 3,9,-27,-81,243,
• 9/33
• -27/9-3
• -81/-273
• 243/-81-3
• Not geometric

4
Find the rule for an for the following sequence.

• 2, 4, 8, 16, 32
• 1st, 2nd, 3rd, 4th, 5th
• Think of how to use the common ratio, n and a1,
  to determine
• the term value.

5
Rule for a Geometric Sequence

• ana1rn-1

• Example 1 Write a rule for the nth term of the
  sequence 5, 20, 80, 320, . Then find a8.
• First, find r.
• r 20/5 4
• an5(4)n-1

a85(4)8-1 a85(4)7 a85(16,384) A881,920
6
Example 2 One term of a geometric sequence is
a43. The common ratio is r3. Write a rule for
the nth term.

• Use ana1rn-1
• 3a1(3)4-1
• 3a1(3)3
• 3a1(27)
• 1/9a1
• ana1rn-1
• an(1/9)(3)n-1

7
Ex 3 Two terms of a geometric sequence are a2-4
and a6-1024. Write a rule for the nth term.

• Write 2 equations, one for each given term.
• a2a1r2-1 OR -4a1r
• a6a1r6-1 OR -1024a1r5
• Use these 2 equations substitution to solve for
  a1 r.
• -4/ra1
• -1024(-4/r)r5
• -1024-4r4
• 256r4
• 4r -4r

If r4, then a1-1. an(-1)(4)n-1
If r-4, then a11. an(1)(-4)n-1 an(-4)n-1
Both Work!
8
Formula for the Sum of a Finite Geometric Series
n of terms a1 1st term r common ratio
9
Example 4 Consider the geometric series
421½ .

• Find the sum of the first 10 terms.

• Find n such that Sn31/4.

10
log232n
11
Looking at infinite series, what happens to the
sum as n approaches infinity in each case?
3 9 27 81, . 3n
27 9 3, 1 1/3 . (1/3)n
Notice, if and
thus the sum does not exist.
12
Looking at infinite series, what happens to the
sum as n approaches infinity if ?
So what if
13
Sum of a Infinite Geometric Series when
n of terms a1 1st term r common ratio
14
Ex 5 Find the Sum of the infinite series
a) 1 1.5 2.25 3.375
Sum DNE since r 1.5 and is gt 1
b) 9 6 4 8/3
r 2/3 and is lt 1 so we use the formula
15
H Dub
9-3 Pg.669 3-42 (3n), 53-55, 73-75, 79-81