Geometric Sequence

1
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).

2
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

3
Rule for a Geometric Sequence

• ana1rn-1

• Example Write a rule for the nth term of the
  sequence 5, 2, 0.8, 0.32, . Then find a8.
• First, find r.
• r 2/5 .4
• an5(.4)n-1

a85(.4)8-1 a85(.4)7 a85(.0016384) a8.008192
4
Example One term of a geometric sequence is
a43. The common ratio is r3. Write a rule for
the nth term. Then graph the sequence.

• If a43, then when n4, an3.
• Use ana1rn-1
• 3a1(3)4-1
• 3a1(3)3
• 3a1(27)
• 1/9a1
• ana1rn-1
• an(1/9)(3)n-1

• To graph, graph the points of the form (n,an).
• Such as, (1,1/9), (2,1/3), (3,1), (4,3),

5
Example 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!
6
Formula for the Sum of a Finite Geometric Series
n of terms a1 1st term r common ratio
7
Example Consider the geometric series 421½ .

• Find the sum of the first 10 terms.

• Find n such that Sn31/4.

8
log232n