1. 11.3 Geometric
Sequences
Day Two
(Long Day of Notes ... Get Started Quickly)
Acts 20:35 "In all things I have shown you that by
working hard in this way we must help the weak and
remember the words of the Lord Jesus, how he himself
said, ‘It is more blessed to give than to receive.’"

2. Partial Sums of Geometric Sequences

3. Partial Sums of Geometric Sequences
Consider:
2 3 n−1
Sn = a + ar + ar + ar +K + ar

4. Partial Sums of Geometric Sequences
Consider:
2 3 n−1
Sn = a + ar + ar + ar +K + ar
2 3 n−1 n
rSn = ar + ar + ar +K + ar + ar

5. Partial Sums of Geometric Sequences
Consider:
2 3 n−1
Sn = a + ar + ar + ar +K + ar
2 3 n−1 n
− rSn = ar + ar + ar +K + ar + ar

6. Partial Sums of Geometric Sequences
Consider:
2 3 n−1
Sn = a + ar + ar + ar +K + ar
2 3 n−1 n
− rSn = ar + ar + ar +K + ar + ar
n
Sn − rSn = a − ar

7. Partial Sums of Geometric Sequences
Consider:
2 3 n−1
Sn = a + ar + ar + ar +K + ar
2 3 n−1 n
− rSn = ar + ar + ar +K + ar + ar
n
Sn − rSn = a − ar
Sn (1− r ) = a (1− r n
)

8. Partial Sums of Geometric Sequences
Consider:
2 3 n−1
Sn = a + ar + ar + ar +K + ar
2 3 n−1 n
− rSn = ar + ar + ar +K + ar + ar
n
Sn − rSn = a − ar
Sn (1− r ) = a (1− r ) n
a (1− r ) n
Sn = r ≠1
(1− r )

9. Partial Sums of Geometric Sequences
Consider:
2 3 n−1
Sn = a + ar + ar + ar +K + ar
2 3 n−1 n
− rSn = ar + ar + ar +K + ar + ar
n
Sn − rSn = a − ar
Sn (1− r ) = a (1− r ) n
a (1− r ) n
Sn = r ≠1
(1− r )
you need the first term (a) and the
common ratio (r)

10. 1) Groups: Find the sum of the first
seventeen terms of the Geometric
Sequence given by
1
36, 6, 1, , K
6

11. 1) Groups: Find the sum of the first
seventeen terms of the Geometric
Sequence given by
1
36, 6, 1, , K
6
1
a1 = 36 r=
6
17
⎛ ⎛ 1 ⎞ ⎞
1− ⎜ ⎟
⎜ ⎝ 6 ⎠ ⎟
S17 = 36 ⎜ ⎟ = 43.2
⎜ 1− ⎛ 1 ⎞ ⎟
⎜ ⎜ ⎟
⎝ 6 ⎠ ⎟
⎝ ⎠

12. 4 k
⎛ 2 ⎞
2) Together let’s find ∑ 7 ⎜ − ⎟
k=1
⎝ 3 ⎠

13. 4 k
⎛ 2 ⎞
2) Together let’s find ∑ 7 ⎜ − ⎟
k=1
⎝ 3 ⎠
14
a1 = −
3

14. 4 k
⎛ 2 ⎞
2) Together let’s find ∑ 7 ⎜ − ⎟
k=1
⎝ 3 ⎠
14 2
a1 = − r=−
3 3

15. 4 k
⎛ 2 ⎞
2) Together let’s find ∑ 7 ⎜ − ⎟
k=1
⎝ 3 ⎠
14 2
a1 = − r=−
3 3
⎛ ⎛ 2 ⎞ 4 ⎞
14 ⎜ 1− ⎜ − 3 ⎟ ⎟
⎝ ⎠
S4 = − ⎜ ⎟
3 ⎜ ⎛ 2 ⎞ ⎟
⎜ 1− ⎜ − ⎟ ⎟
⎝ ⎝ 3 ⎠ ⎠

16. 4 k
⎛ 2 ⎞
2) Together let’s find ∑ 7 ⎜ − ⎟
k=1
⎝ 3 ⎠
14 2
a1 = − r=−
3 3
⎛ ⎛ 2 ⎞ 4 ⎞
14 ⎜ 1− ⎜ − 3 ⎟ ⎟
⎝ ⎠
S4 = − ⎜ ⎟
3 ⎜ ⎛ 2 ⎞ ⎟
⎜ 1− ⎜ − ⎟ ⎟
⎝ ⎝ 3 ⎠ ⎠
182
S4 = −2.24691358 → −
81

17. A partial sum is when we add a finite
number of terms of a sequence.

18. A partial sum is when we add a finite
number of terms of a sequence.
An Infinite Series is when we add an
infinite number of terms of a sequence.

19. A partial sum is when we add a finite
number of terms of a sequence.
An Infinite Series is when we add an
infinite number of terms of a sequence.
note: we will be using this math fact
1
→ 0 when n → ∞
n

20. A partial sum is when we add a finite
number of terms of a sequence.
An Infinite Series is when we add an
infinite number of terms of a sequence.
note: we will be using this math fact
1
→ 0 when n → ∞
n
or
k
→ 0 when n → ∞
n

21. Sums of Infinite Series

22. Sums of Infinite Series
Can the sum of an infinite number of
terms converge to a limiting value?

23. Sums of Infinite Series
Can the sum of an infinite number of
terms converge to a limiting value?
Consider: Lance is 27 feet from a wall.
If he goes 1/3 of the way to the wall on
each “trip”,

24. Sums of Infinite Series
Can the sum of an infinite number of
terms converge to a limiting value?
Consider: Lance is 27 feet from a wall.
If he goes 1/3 of the way to the wall on
each “trip”,
a) how many trips will he take?

25. Sums of Infinite Series
Can the sum of an infinite number of
terms converge to a limiting value?
Consider: Lance is 27 feet from a wall.
If he goes 1/3 of the way to the wall on
each “trip”,
a) how many trips will he take?
b) how long will Lance be “trippin”

26. Sums of Infinite Series
Can the sum of an infinite number of
terms converge to a limiting value?
Consider: Lance is 27 feet from a wall.
If he goes 1/3 of the way to the wall on
each “trip”,
a) how many trips will he take?
b) how long will Lance be “trippin”
c) how far will Lance have gone?

27. Sums of Infinite Series
Can the sum of an infinite number of
terms converge to a limiting value?
Consider: Lance is 27 feet from a wall.
If he goes 1/3 of the way to the wall on
each “trip”,
a) how many trips will he take?
b) how long will Lance be “trippin”
c) how far will Lance have gone?
a) ∞ # trips is like the # of terms

28. Sums of Infinite Series
Can the sum of an infinite number of
terms converge to a limiting value?
Consider: Lance is 27 feet from a wall.
If he goes 1/3 of the way to the wall on
each “trip”,
a) how many trips will he take?
b) how long will Lance be “trippin”
c) how far will Lance have gone?
a) ∞ # trips is like the # of terms
b) forever - never reaches the wall

29. Sums of Infinite Series
Can the sum of an infinite number of
terms converge to a limiting value?
Consider: Lance is 27 feet from a wall.
If he goes 1/3 of the way to the wall on
each “trip”,
a) how many trips will he take?
b) how long will Lance be “trippin”
c) how far will Lance have gone?
a) ∞ # trips is like the # of terms
b) forever - never reaches the wall
c) ≈ 27 ft. this is the limiting value of
8
9 + 6 + 4 + +K
3

30. Consider the Geometric Series:
2 3
a + ar + ar + ar +K

31. Consider the Geometric Series:
2 3
a + ar + ar + ar +K
n
if r > 1, then r → ∞ as n → ∞

32. Consider the Geometric Series:
2 3
a + ar + ar + ar +K
n
if r > 1, then r → ∞ as n → ∞
a (1− r n
) a (1− ∞ )
so Sn = becomes Sn = or Sn diverges
(1− r ) (1− r )

33. Consider the Geometric Series:
2 3
a + ar + ar + ar +K
n
if r > 1, then r → ∞ as n → ∞
a (1− r n
) a (1− ∞ )
so Sn = becomes Sn = or Sn diverges
(1− r ) (1− r )
n
if r < 1, then r → 0 as n → ∞

34. Consider the Geometric Series:
2 3
a + ar + ar + ar +K
n
if r > 1, then r → ∞ as n → ∞
a (1− r n
) a (1− ∞ )
so Sn = becomes Sn = or Sn diverges
(1− r ) (1− r )
n
if r < 1, then r → 0 as n → ∞
a (1− r n
) a (1− 0 )
so Sn = becomes Sn = or Sn converges
(1− r ) (1− r )
a
to
1− r

35. Consider the Geometric Series:
2 3
a + ar + ar + ar +K
n
if r > 1, then r → ∞ as n → ∞
a (1− r n
) a (1− ∞ )
so Sn = becomes Sn = or Sn diverges
(1− r ) (1− r )
n
if r < 1, then r → 0 as n → ∞
a (1− r n
) a (1− 0 )
so Sn = becomes Sn = or Sn converges
(1− r ) (1− r )
a
to
1− r

36. Together let’s determine if
10 10 10
10 + + 2 + 3 +K
7 7 7
converges or diverges. If it converges,
show the value to which it converges.

37. Together let’s determine if
10 10 10
10 + + 2 + 3 +K
7 7 7
converges or diverges. If it converges,
show the value to which it converges.
This is a geometric series with
1
a1 = 10 and r=
7

38. Together let’s determine if
10 10 10
10 + + 2 + 3 +K
7 7 7
converges or diverges. If it converges,
show the value to which it converges.
This is a geometric series with
1
a1 = 10 and r=
7
Since r < 1 this will converge to

39. Together let’s determine if
10 10 10
10 + + 2 + 3 +K
7 7 7
converges or diverges. If it converges,
show the value to which it converges.
This is a geometric series with
1
a1 = 10 and r=
7
Since r < 1 this will converge to
a 10 10 70
= = =
1− r 1− 1 6 6
7 7

40. A superball is dropped from a height
of 8 feet. Each time it hits the ground
it bounces back to a height 70% of the
distance it fell. Find the total
vertical distance it traveled.

41. A superball is dropped from a height
of 8 feet. Each time it hits the ground
it bounces back to a height 70% of the
distance it fell. Find the total
vertical distance it traveled.
(A problem similar to this will be on
your test and your final exam.)

42. A superball is dropped from a height
of 8 feet. Each time it hits the ground
it bounces back to a height 70% of the
distance it fell. Find the total
vertical distance it traveled.
(A problem similar to this will be on
your test and your final exam.)

43. A superball is dropped from a height
of 8 feet. Each time it hits the ground
it bounces back to a height 70% of the
distance it fell. Find the total
vertical distance it traveled.
(A problem similar to this will be on
your test and your final exam.)
notice that the
ball falls 8 ft.
once and the
other distances
twice.

44. 2 3
d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K

45. 2 3
d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K
2 3
d = 8 + 16 (.7 ) + 16 (.7 ) + 16 (.7 ) +K

46. 2 3
d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K
2 3
d = 8 + 16 (.7 ) + 16 (.7 ) + 16 (.7 ) +K
The green part is an infinite
geometric series

47. 2 3
d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K
2 3
d = 8 + 16 (.7 ) + 16 (.7 ) + 16 (.7 ) +K
The green part is an infinite
geometric series
r = .7 ∴ converges a = 16 (.7 ) = 11.2

48. 2 3
d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K
2 3
d = 8 + 16 (.7 ) + 16 (.7 ) + 16 (.7 ) +K
The green part is an infinite
geometric series
r = .7 ∴ converges a = 16 (.7 ) = 11.2
a 11.2 1
= = 37
1− r 1− .7 3

49. 2 3
d = 8 + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) + 2 (.7 ) ( 8 ) +K
2 3
d = 8 + 16 (.7 ) + 16 (.7 ) + 16 (.7 ) +K
The green part is an infinite
geometric series
r = .7 ∴ converges a = 16 (.7 ) = 11.2
a 11.2 1
= = 37
1− r 1− .7 3
1 1
d = 8 + 37 = 45 ft.
3 3

50. HW #8
Quiz Tomorrow!
“Give a man a fish and you feed him for a day.
Teach a man to fish and you feed him for a lifetime.”
chinese proverb