12.3 – Analyze Geometric Sequences and Series

Geometric Sequence:

Ratio of any term to the previous term is constant

Common Ratio:

Ratio each term is increasing by

1. Tell whether the sequence is geometric.

4, 10, 18, 28, 40, …

52

95

14 9

10 7

No

2. Tell whether the sequence is geometric.

625, 125, 25, 5, 1, …

15

15

15

15

Yes

3. Tell whether the sequence is geometric.

–4, 8, –16, 32, –64, …

–2

Yes

–2 –2 –2

Rule for a Geometric Sequence:

The nth term of a geometric sequence with first term a1 and common ratio r is given by:

11

nna a r

4. Write a rule for the nth term of the sequence. Then find a7.

4, 20, 100, 500, ….

11

nna a r

a1 = 4

r = 5

14 5

nna

a7 = 4(5)7–1

a7 = 4(5)6

a7 = 4(15625)

a7 = 62500

5. Write a rule for the nth term of the sequence. Then find a7.

152, –76, 38, –19, …

11

nna a r

a1 = 152

r =-1 2

11

1522

n

na

a7 = 152(-1/2)7–1

a7 = 152(-1/2)6

a7 = 152(1/64)

a7 = 19/8

6. Write a rule for the nth term of the geometric sequence, given: a4 = 12, r = 2

11

nna a r

12 = a1(2)4 – 1

12 = a1(2)3

12 = 8a1

3/2 = a1

132

2n

na

7. Write a rule for the nth term of the geometric sequence, given: a6 = –96, r = 2/3

11

nna a r

12

7293

n

na

6 1

12

963

a

5

12

963

a

132

96243

a 243

32

1729 a

8. Write a rule for the nth term of the geometric sequence, given: a3 = –48, a6 = 3072 1

1n

na a r

3 1148 a r

6 113072 a r

2148 a r 5

13072 a r

12

48a

r

52

483072 r

r

r3

33072 48r364 r

4 r

148

16a

13 a

13 4

nna

9. Write a rule for the nth term of the geometric sequence, given: a2 = –12, a4 = –3 1

1n

na a r

2 1112 a r

4 113 a r

112 a r 3

13 a r

112

ar

3123 r

r

r2

23 12r 21

4r

1

2r

112

0.5a

124 a

11

242

n

na

Sum of a Finite Geometric Series:

The sum of the first n terms of a geometric series with common ratio r 1 is:

11

1

n

nr

S ar

11

1

n

nr

S ar

10. Find the sum of the geometric series.

16

1

1

4 3i

i

a1 = 4(3)1-1 =

r = 3

4(1) = 4

n = 16

161 34

1 3nS 1 43046721

41 3nS

4 21523360

86,093,440S

11

1

n

nr

S ar

11. Find the sum of the geometric series.7

0

112

2

i

i

a1 = 12(-1/2)0 =

r = -1/2

12(1) = 12

n = 8

81 0.5

121 0.5nS

1 0.00390625

121.5nS

8512

128

255

32S