Geometric Sequences

Section 3.2.1

Vocabulary

Geometric Sequence: A sequence in which the ratio of any term to the previous term is constant.

Common Ratio: The constant ratio between consecutive terms of a geometric sequence, denoted by r.

Investigation 1: Recall: An arithmetic sequence is a sequence in which the difference between two consecutive terms is constant.

The constant difference between terms of an arithmetic sequence is denoted d and the explicit formula to find the nth term of a sequence is: an = a1 + d(n – 1).

1. Identify the next three terms of the arithmetic sequence, then write the explicit formula for the

sequence: 3, 7, 11, 15, an = 3 + 4(n – 1) or an = 4n – 1

2. Use the formula from example #1 to find the 27th term of the sequence.

a27 = 3 + 4(27 – 1) =

19, 23, 27, . . .

107

In an arithmetic sequence, the terms are found by adding a constant amount to the preceding term. In a geometric sequence, the terms are found by multiplying each term after the first by a constant amount. This constant multiplier is called the common ratio and is denoted r.

For each geometric sequence, identify the common ratio, r.3. 2, 6, 18, 54, 162, . . .

4. 5, 50, 500, 5000, . . .

5. 3, , , , . . .

6. -4, 24, -144, 864, -5184, . . .

3

2

3

43

8

r = 3

r = 10

r = ½

r = -6

Tell whether the sequences is arithmetic, geometric or neither. For arithmetic sequences, give the common difference. For geometric sequences, give the common ratio.  7. 5, 10, 15, 20, 25, …. 8. 1, 1, 2, 3, 5, 8, 13, 21, …

9. 1, -4, 16, -64, 256, …

10. 512, 256, 128, 64, 32, …

arithmetic; d = 5

neither

geometric; r = -4

geometric; r = ½

Check for Understanding:  11. Find the first four terms of a geometric sequence in which a1 = 5 and r = -3.

_____ , _____ , _____ , _____.  

12. Find the missing term in the geometric sequence: -7, _______ , -28, 56, _______ , . . .

5 -15 45 -135

× -3 × -3 × -3

56 ÷ -28 = -2So, r = -2

× -2

14

× -2

-112

Investigation 2: The explicit formula used to find the nth term of a geometric sequence with the first term a1 and the common ratio r is given by: an = a1∙ rn-1

Write a rule for the nth term of the sequence given. Then find a10.  

  

13. 1, 6, 36, 216, 1296, …

Rule: an = 1∙6n-1

a10 = 1∙610-1 = 10077696

  14. 14, 28, 56, 112, …

Rule: an = 14∙2n-1

a10 = 14∙210-1 = 7168

Check for Understanding: 15. If a5 = 324 and r = -3, write the explicit formula for the geometric sequence and find a10.

_____ , _____ , _____ , _____, 324 

Rule: an = 4∙(-3)n-1

a10 = 4∙(-3)10-1 = -78732 

÷ -3

-108

÷ -3

36

÷ -3

-12

÷ -3

4

5 1

1324 3a

1324 81a

14 a

OR

16. If a3 = 18 and r = 3 write the explicit formula for the geometric sequence and find a10.

Rule: an = 2∙(3)n-1

a10 = 2∙(3)10-1 = 39366

3 1

118 3a

118 9a

12 a

20. If r = 2 and a1 = 1 for a geometric sequence, a. Write a rule for the nth term of the sequence.

b. Graph the first five terms of the sequence. (1, 1), (2, 2), (3, 4), (4, 8), (5, 16)

c. What kind of graph does this represent? exponential

11 2

n

na