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.

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.
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.
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.
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.
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.
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.

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.
S15 = a1(1 – r15)

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.
S15 = a1(1 – r15)
S15 = 2(1 – r15)

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.
S15 = a1(1 – r15)
S15 = 2(1 – r15)
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.
S15 = a1(1 – r15)
S15 = 2(1 – r15)
S15 = 2(1 – 215)
1 – 2
S15 = 2(1 – 215) = 2(1 – 32,768)
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.
S15 = a1(1 – r15)
S15 = 2(1 – r15)
S15 = 2(1 – 215)
1 – 2
S15 = 2(1 – 215) = 2(1 – 32,768) = -65,534
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.
S15 = a1(1 – r15)
S15 = 2(1 – r15)
S15 = 2(1 – 215)
1 – 2
S15 = 2(1 – 215) = 2(1 – 32,768) = -65,534 = 65,534
1 –