11.3: Geometric Sequences

1
11.3 Geometric Sequences Series
2

• nth Term of an Geometric Sequence
• an a1r(n 1)

3

• nth Term of an Geometric Sequence
• an a1r(n 1)
• Ex. 2 Determine the following
• a) Find the eighth term of a geometric
  sequence for which a1 -3 and r -2.

4

• nth Term of an Geometric Sequence
• an a1r(n 1)
• Ex. 2 Determine the following
• a) Find the eighth term of a geometric
  sequence for which a1 -3 and r -2.
• an a1r(n 1)

5

• nth Term of an Geometric Sequence
• an a1r(n 1)
• Ex. 2 Determine the following
• a) Find the eighth term of a geometric
  sequence for which a1 -3 and r -2.
• an a1r(n 1)
• a8 a1r(8 1)

6

• nth Term of an Geometric Sequence
• an a1r(n 1)
• Ex. 2 Determine the following
• a) Find the eighth term of a geometric
  sequence for which a1 -3 and r -2.
• an a1r(n 1)
• a8 a1r(8 1)
• a8 (-3)r(8 1)

7

• nth Term of an Geometric Sequence
• an a1r(n 1)
• Ex. 2 Determine the following
• a) Find the eighth term of a geometric
  sequence for which a1 -3 and r -2.
• an a1r(n 1)
• a8 a1r(8 1)
• a8 (-3)r(8 1)
• a8 (-3)(-2)(8 1)

8

• nth Term of an Geometric Sequence
• an a1r(n 1)
• Ex. 2 Determine the following
• a) Find the eighth term of a geometric
  sequence for which a1 -3 and r -2.
• an a1r(n 1)
• a8 a1r(8 1)
• a8 (-3)r(8 1)
• a8 (-3)(-2)(8 1)
• a8 (-3)(-27)

9

• nth Term of an Geometric Sequence
• an a1r(n 1)
• Ex. 2 Determine the following
• a) Find the eighth term of a geometric
  sequence for which a1 -3 and r -2.
• an a1r(n 1)
• a8 a1r(8 1)
• a8 (-3)r(8 1)
• a8 (-3)(-2)(8 1)
• a8 (-3)(-27)
• a8 (-3)(-128)

10

• nth Term of an Geometric Sequence
• an a1r(n 1)
• Ex. 2 Determine the following
• a) Find the eighth term of a geometric
  sequence for which a1 -3 and r -2.
• an a1r(n 1)
• a8 a1r(8 1)
• a8 (-3)r(8 1)
• a8 (-3)(-2)(8 1)
• a8 (-3)(-27)
• a8 (-3)(-128)
• a8 384

11

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,

12

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• 12
• 3

13

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• 4

14

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4

15

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)

16

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)r(n 1)

17

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)(4)(n 1)

18

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)4n 1

19

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)4n 1
• c) Find the tenth term of a geometric sequence
  for which a4 108 and r 3.

20

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)4n 1
• c) Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1

21

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)4n 1
• c) Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 an a1r(n 1)

22

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)4n 1
• c) Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)

23

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)4n 1
• c) Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)

24

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)4n 1
• c) Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)

25

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)4n 1
• c) Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)
• 108 a133

26

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)4n 1
• c) Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)
• 108 a133
• 108 27a1

27

• b) Write an equation for the nth term of the
  geometric sequence 3, 12, 48, 192,
• r 4
• an a1r(n 1)
• an (3)4n 1
• c) Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)
• 108 a133
• 108 27a1
• 4 a1

28

• c)Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)
• 108 a133
• 108 27a1
• 4 a1
• 2. Now use a1 4 and r 3 to find a10

29

• c)Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)
• 108 a133
• 108 27a1
• 4 a1
• 2. Now use a1 4 and r 3 to find a10
• an a1r(n 1)

30

• c)Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)
• 108 a133
• 108 27a1
• 4 a1
• 2. Now use a1 4 and r 3 to find a10
• an a1r(n 1)
• a10 a1r(10 1)

31

• c)Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)
• 108 a133
• 108 27a1
• 4 a1
• 2. Now use a1 4 and r 3 to find a10
• an a1r(n 1)
• a10 a1r(10 1)
• a10 (4)r(10 1)

32

• c)Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)
• 108 a133
• 108 27a1
• 4 a1
• 2. Now use a1 4 and r 3 to find a10
• an a1r(n 1)
• a10 a1r(10 1)
• a10 (4)r(10 1)
• a10 (4)(3)(10 1)

33

• c)Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)
• 108 a133
• 108 27a1
• 4 a1
• 2. Now use a1 4 and r 3 to find a10
• an a1r(n 1)
• a10 a1r(10 1)
• a10 (4)r(10 1)
• a10 (4)(3)(10 1)
• a10 (4)(39)

34

• c)Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)
• 108 a133
• 108 27a1
• 4 a1
• 2. Now use a1 4 and r 3 to find a10
• an a1r(n 1)
• a10 a1r(10 1)
• a10 (4)r(10 1)
• a10 (4)(3)(10 1)
• a10 (4)(39)
• a10 (4)(19,683)

35

• c)Find the tenth term of a geometric sequence
  for which a4 108 and r 3.
• 1. Find a1 a4 a1r(4 1)
• 108 a1r(4 1)
• 108 a13(4 1)
• 108 a133
• 108 27a1
• 4 a1
• 2. Now use a1 4 and r 3 to find a10
• an a1r(n 1)
• a10 a1r(10 1)
• a10 (4)r(10 1)
• a10 (4)(3)(10 1)
• a10 (4)(39)
• a10 (4)(19,683)
• a10 78,732

36

• Sum of a Geometric Series
• The sum Sn of the first n terms of a geometric
  series is given by the following
• Sn a1(1 r n)
• 1 r

37

• Sum of a Geometric Series
• The sum Sn of the first n terms of a geometric
  series is given by the following
• Sn a1(1 r n)
• 1 r
• Ex. 3 Find the sum of the first 15 terms of the
  geometric sequence in which a1 2 and r 2.

38

• Sum of a Geometric Series
• The sum Sn of the first n terms of a geometric
  series is given by the following
• Sn a1(1 r n)
• 1 r
• Ex. 3 Find the sum of the first 15 terms of the
  geometric sequence in which a1 2 and r 2.
• Sn a1(1 r n)
• 1 r

39

• Sum of a Geometric Series
• The sum Sn of the first n terms of a geometric
  series is given by the following
• Sn a1(1 r n)
• 1 r
• Ex. 3 Find the sum of the first 15 terms of the
  geometric sequence in which a1 2 and r 2.
• Sn a1(1 r n)
• 1 r
• S15 a1(1 r15)
• 1 r

40

• Sum of a Geometric Series
• The sum Sn of the first n terms of a geometric
  series is given by the following
• Sn a1(1 r n)
• 1 r
• Ex. 3 Find the sum of the first 15 terms of the
  geometric sequence in which a1 2 and r 2.
• Sn a1(1 r n)
• 1 r
• S15 a1(1 r15)
• 1 r
• S15 2(1 r15)
• 1 r

41

• Sum of a Geometric Series
• The sum Sn of the first n terms of a geometric
  series is given by the following
• Sn a1(1 r n)
• 1 r
• Ex. 3 Find the sum of the first 15 terms of the
  geometric sequence in which a1 2 and r 2.
• Sn a1(1 r n)
• 1 r
• S15 a1(1 r15)
• 1 r
• S15 2(1 r15)
• 1 r
• S15 2(1 215)
• 1 2

42

• Sum of a Geometric Series
• The sum Sn of the first n terms of a geometric
  series is given by the following
• Sn a1(1 r n)
• 1 r
• Ex. 3 Find the sum of the first 15 terms of the
  geometric sequence in which a1 2 and r 2.
• Sn a1(1 r n)
• 1 r
• S15 a1(1 r15)
• 1 r
• S15 2(1 r15)
• 1 r
• S15 2(1 215)
• 1 2
• S15 2(1 215) 2(1 32,768)
• 1 2 -1

43

• Sum of a Geometric Series
• The sum Sn of the first n terms of a geometric
  series is given by the following
• Sn a1(1 r n)
• 1 r
• Ex. 3 Find the sum of the first 15 terms of the
  geometric sequence in which a1 2 and r 2.
• Sn a1(1 r n)
• 1 r
• S15 a1(1 r15)
• 1 r
• S15 2(1 r15)
• 1 r
• S15 2(1 215)
• 1 2
• S15 2(1 215) 2(1 32,768) -65,534
• 1 2 -1
  -1

44

• Sum of a Geometric Series
• The sum Sn of the first n terms of a geometric
  series is given by the following
• Sn a1(1 r n)
• 1 r
• Ex. 3 Find the sum of the first 15 terms of the
  geometric sequence in which a1 2 and r 2.
• Sn a1(1 r n)
• 1 r
• S15 a1(1 r15)
• 1 r
• S15 2(1 r15)
• 1 r
• S15 2(1 215)
• 1 2
• S15 2(1 215) 2(1 32,768) -65,534
  65,534
• 1 2 -1
  -1