Deck 10: Sequences and Infinite Series

Question 1

Write the first four elements of the sequence.

-

A)

B)

C)

D)

Answer 1

Question 2

Write the first four elements of the sequence.

-

A)

B)

C)

D)

Answer 2

Question 3

Write the first four elements of the sequence.

-

A)

B)

C)

D)

Answer 3

Question 4

Write the first four elements of the sequence.

-

A)

B)

C)

D)

Question 5

Write the first four elements of the sequence.

-sin (n $\pi$ )

A) 1, 0, -1, 0
B) 0, 1, 0, -1
C) 1, 1, 1, 1
D) 0, 0, 0, 0

Question 6

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.

- = 1, = 2

A) 2, 3, 4, 5, 6
B) 1, 2, 4, 8, 16
C) 2, 4, 8, 16, 32
D) 1, 3, 5, 7, 9

Question 7

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.

- = 1, =

A) 1, 1, 1, 1, 1
B) 1, 2, 4, 8, 16, 32
C) 1, 3, 5, 7, 9
D) 1, 2, 4, 8, 16

Question 8

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.

- = 1, = + 5

A) 1, 6, 11, 16, 21, 26
B) 1, 6, 11, 16, 21
C) 6, 11, 16, 21, 26
D) 1, 5, 25, 125, 625, 3125

Question 9

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.

- = 4, = -

A) 4, -16, 64, -256, 1024
B) 4, 0, -4, -8, -12
C) 4, -4, 4, -4, 4
D) -4, 4, -4, 4, -4

Question 10

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.

- = 6, =

A) -6, 6, -6, 6, -6
B) 6, -6, 6, -6, 6
C) 6, -6, -6, 6, 6
D) 6, 6, -6, -6, 6

Question 11

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.

- = 3, =

A)

B)

C)

D)

Question 12

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.

- = 1, =

A)

B)

C)

D)

Question 13

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.

- = 1, =

A)

B)

C)

D)

Question 14

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.

-

A) 1, 4, 5, 10, 15
B) 1, 4, 5, 9, 14
C) 1, 1, 2, 3, 5
D) 1, 4, 5, 6, 7

Question 15

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.

-

A) 1, 3, 2, -1, -3
B) 1, -3, 4, -5, 6
C) 1, 3, 2, 1, 0
D) 1, -1, 2, -3, 5

Question 16

Find a formula for the nth term of the sequence.

-7, 8, 9, 10, 11 (integers beginning with 7)

A) = n + 6
B) = n + 7
C) = n - 7
D) = n + 8

Question 17

Find a formula for the nth term of the sequence.

--5, -4, -3, -2, -1 (integers beginning with -5)

A) = n - 6
B) = n - 5
C) = n + 5
D) = n - 4

Question 18

Find a formula for the nth term of the sequence.

-6, -6, 6, -6, 6 ( 6's with alternating signs)

A) = 6
B) = 6
C) = 6
D) = 6

Question 19

Find a formula for the nth term of the sequence.

-1, - , , - , (reciprocals of squares with alternating signs)

A) =
B) =
C) =
D) =

Question 20

Find a formula for the nth term of the sequence.

-6, 8, 10, 12, 14 (every other integer starting with 6)

A) = 2n + 5
B) = 2n + 4
C) = n + 11
D) = n + 10

Question 21

Find a formula for the nth term of the sequence.

-0, 2, 0, 2, 0 (alternating 0's and 2's)

A) = 1 +
B) = 1 +
C) = 1 +
D) = 1 -

Question 22

Find a formula for the nth term of the sequence.

-0, , 0, , 0 (alternating 0's and 's)

A) =
B) =
C) =
D) =

Question 23

Find a formula for the nth term of the sequence.

-0, 0, 2, 2, 0, 0, 2, 2 (alternating 0's and 2's in pairs)

A) = 1 -
B) =
C) = 1 +
D) = 1 +

Question 24

Find a formula for the nth term of the sequence.

-0, 2, 2, 2, 0, 2, 2, 2 (0, 2, 2, 2 repeated)

A) = 1 +
B) = 1 +
C) = 1 +
D) = 1 +

Question 25

Find a formula for the nth term of the sequence.

-0, -1, 0, 1, 0, -1, 0, 1 (0, -1, 0, 1 repeated)

A) = sin
B) = sin(n $\pi$ )
C) = cos(n $\pi$ )
D) = cos

Question 26

Find the limit of the sequence if it converges; otherwise indicate divergence.

- = 4 +

A) 4.5
B) 4
C) 5
D) Diverges

Question 27

Find the limit of the sequence if it converges; otherwise indicate divergence.

- =

A) -10
B)
C)
D) Diverges

Question 28

Find the limit of the sequence if it converges; otherwise indicate divergence.

- =

A) 3
B) ±3
C) 0
D) Diverges

Question 29

Find the limit of the sequence if it converges; otherwise indicate divergence.

- =

A) 2
B) 0
C) 1
D) Diverges

Question 30

Find the limit of the sequence if it converges; otherwise indicate divergence.

- =

A) -9
B)
C)
D) Diverges

Question 31

Find the limit of the sequence if it converges; otherwise indicate divergence.

- =

A)
B)
C) 1
D) Diverges

Question 32

Find the limit of the sequence if it converges; otherwise indicate divergence.

- =

A)
B) 1
C) 0
D) Diverges

Question 33

Find the limit of the sequence if it converges; otherwise indicate divergence.

- = 1 + +

A) 1
B) 3
C) 0
D) Diverges

Question 34

Assume that the sequence converges and find its limit.

- = 5, =

A) 6
B) 8
C) 4
D) 5

Question 35

Assume that the sequence converges and find its limit.

- = -2, =

A) -6
B) 8
C) -2
D) 4

Question 36

Assume that the sequence converges and find its limit.

- = 4, =

A) -4
B) 4
C) 1
D) 16

Question 37

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.

-3 + + + ... + + ...

A)
; 4
B)
;
C)
;
D)
; 4

Question 38

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.

-7 + 63 + 567 + ... + 7 . + ...

A)
;
B)
;
C)
; series diverges
D)
; series diverges

Question 39

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.

-3 - + - + ... + + ...

A)
; 4
B)
; 4
C)
;
D)
;

Question 40

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.

-3 - 21 + 147 - 1029 + ... + 3 . + ...

A)
;
B)
; series diverges
C)
; series diverges
D)
;

Question 41

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.

- + + + ... + + ...

A)
; 2
B)
; 2
C)
; 2
D)
; 2

Question 42

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.

- + + + ... + + ...

A)
;
B)
; 3
C)
;
D)
; 3

Question 43

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.

- + + + ... + + ...

A)
; 3
B)
; 3
C)
;
D)
;

Question 44

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.

- + + + ... + + ...

A)
;
B)
;
C)
;
D)
;

Question 45

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.

- + + + ... + + ...

A)
; 8
B)
; 8
C)
; 8
D)
; 8

Question 46

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.

- + + + ... + + ...

A)
; 2
B)
; 2
C)
; 1
D)
; 1

Question 47

Solve the problem.

-A company's annual revenue for the period since 2000 can be modeled by the function where R is in millions of dollars and n = 0 corresponds to 2000. Assuming the model accurately predicts future revenue, find the year in which the revenue first exceeds $4.98 million.

A) 2011
B) 2009
C) 2013
D) 2010

Question 48

Solve the problem.

-At a plant that packages bottled spring water, the water is passed through a sequence of ion-exchange filters to reduce the sodium content prior to bottling. Each filter removes 70% of the sodium present in the water passing through it. Determine the number of filters that must be used to reduce the sodium concentration from 30 parts-per-million to 7.20 parts-per-million.

A) 4 filters
B) 6 filters
C) 7 filters
D) 5 filters

Question 49

Find the limit of the sequence or determine that the limit does not exist.

- = ln( 9n + 9) - ln( 2n - 7)

A) does not exist
B) ln
C) ln 7
D) ln

Question 50

Find the limit of the sequence or determine that the limit does not exist.

- =

A) ln 7
B) does not exist
C)
D) 0

Question 51

Find the limit of the sequence or determine that the limit does not exist.

- =

A) does not exist
B) ln 3
C) 0
D) 1

Question 52

Find the limit of the sequence or determine that the limit does not exist.

- =

A) 1
B) does not exist
C) e
D)

Question 53

Find the limit of the sequence or determine that the limit does not exist.

- =

A) ln 9
B) 0
C) does not exist
D) 1

Question 54

Find the limit of the sequence or determine that the limit does not exist.

- =

A)
B) 0
C) 1
D) does not exist

Question 55

Find the limit of the sequence or determine that the limit does not exist.

- =

A) 1
B) ln 6
C) 0
D) does not exist

Question 56

Find the limit of the sequence or determine that the limit does not exist.

- =

A) 0
B) does not exist
C) 1
D) 5

Question 57

Find the limit of the sequence or determine that the limit does not exist.

- =

A) 0
B) 5
C) does not exist
D)

Question 58

Find the limit of the sequence or determine that the limit does not exist.

- = ln

A) 0
B) 3
C) does not exist
D) ln 3

Question 59

Find the limit of the sequence or determine that the limit does not exist.

- =

A) 0
B) 1
C)
D) does not exist

Question 60

Find the limit of the sequence or determine that the limit does not exist.

- = n -

A)
B)
C) does not exist
D) 0

Question 61

Find the limit of the sequence or determine that the sequence diverges.

-

A) 7
B) The sequence diverges.
C) - 7
D) 0

Question 62

Solve the problem.

-A patient takes 100 mg of medication every 24 hours. 80% of the medication in the blood is eliminated every 24 hours.

a. Let d_n equal the amount of medication (in mg) in the blood stream after n doses, where d₁ = 100. Find a recurrence relation for d_n .
b. Show that (d_n) is monotonic and bounded, and therefore converges.
c. Find the limit of the sequence. What is the physical meaning of this limit?

A) a. d_n + 1 = 0.2 d_n + 100, d₁ d₁ = 100
b. d_n satisfies 0 ≤ d_n ≤ 125 for n ≥ 1 and its terms are increasing in size.
c. 125; in the long run there will be approximately 125 mg of medication in the blood.

B) a. d_n + 1 = 0.8 d_n + 100, d₁ = 100
b. d_n satisfies 0 ≤ d_n ≤ 150 for n ≥ 1 and its terms are increasing in size.
c. 150; in the long run there will be approximately 150 mg of medication in the blood.


C) a. d_n + 1 = 0.2 d_n + 100, d₁ = 100
b. d_n satisfies 0 ≤ d_n ≤ 150 for n ≥ 1 and its terms are increasing in size.
c. 150; in the long run there will be approximately 150 mg of medication in the blood.

D) a. d_n + 1 = 0.8d_n + 100, d₁ = 100
b. d_n satisfies 0 ≤ d_n ≤ 125 for n ≥ 1 and its terms are increasing in size.
c. 125; in the long run there will be approximately 125 mg of medication in the blood.

Question 63

Find the limit of the sequence or state that it diverges.

-

A) 0
B) -1
C) diverges
D) 1

Question 64

Find the limit of the sequence or state that it diverges.

-

A) 0
B) diverges
C) 1
D) -1

Question 65

Determine if the geometric series converges or diverges. If it converges, find its sum.

-1 + + + + + . . .

A) converges, 3
B) diverges
C) converges,
D) converges,

Question 66

Determine if the geometric series converges or diverges. If it converges, find its sum.

-1 + ( -9) + + + + . . .

A) converges, 9
B) converges, 81
C) diverges
D) converges, 90

Question 67

Determine if the geometric series converges or diverges. If it converges, find its sum.

- + + + + + . . .

A) converges,
B) diverges
C) converges,
D) converges,

Question 68

Find the sum of the series.

-

A) 24/7
B) 3/7
C) 8/3
D) 1/3

Question 69

Find the sum of the series.

-

A) 1/2
B) 5/3
C) 1/3
D) 5/2

Question 70

Find the sum of the series.

-

A) 5/2
B) 1/2
C) 1/3
D) 5/3

Question 71

Find the sum of the series.

-

A) 42/5
B) 6
C) 1
D) 7/5

Question 72

Find the sum of the series.

-

A) 23/8
B) 143/56
C) 145/56
D) 127/56

Question 73

Find the sum of the series.

-

A) 1/4
B) 1
C) - 1/4
D) -1

Question 74

Find the sum of the series.

-

A) 5/6
B) - 5/6
C) - 1/3
D) 1/3

Question 75

Find the sum of the series.

-

A) - 33/8
B) 3/10
C) 159/8
D) 123/10

Question 76

Find the sum of the series.

-

A) 12/37
B) 6/37
C) 6/35
D) 12/35

Question 77

Find the sum of the series.

-

A) 65536/3
B) 1/ 786432
C) 262144/3
D) 1/196608

Question 78

Express the number as the ratio of two integers.

-0.77777 . . .

A) 7/99
B) 70/99
C) 7/9
D) 70/999

Question 79

Express the number as the ratio of two integers.

-0.747474 . . .

A) 74/99
B) 20/27
C) 740/99
D) 2/27

Question 80

Express the number as the ratio of two integers.

-0.5303030 . . .

A) 175/33
B) 35/22
C) 35/66
D) 35/33