Deck 7: Sequences; Induction; the Binomial Theorem

Question 1

Write out the first five terms of the sequence.
{2(4n - 3)}

A) 2, 4, 6, 8, 10
B) 1, 5, 9, 13, 17
C) -6, 2, 10, 18, 26
D) 2, 10, 18, 26, 34

Answer 1

D

Question 2

Evaluate the factorial expression.
$\frac { 7 ! } { 9 ! }$

A) 2 !
B) $\frac { 1 } { 72 }$
C) $\frac { 1 } { 2 ! }$
D) 72

Answer 2

B

Question 3

Evaluate the factorial expression.
$$\frac { ( n + 4 ) ! } { n + 4 }$$

A) 1
B) $n + 4$ !
C) $4 !$
D) $( n + 3 ) !$

Answer 3

D

Question 4

Write out the first five terms of the sequence.
{3n - 1}

A) 2, 5, 8, 11, 14
B) 4, 7, 10, 13, 16
C) -2, -5, -8, -11, -14
D) 2, 3, 4, 5, 6

Question 5

Write out the first five terms of the sequence.
{n - 1}

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

Question 6

Evaluate the factorial expression.
$$\frac { n ( n + 6 ) ! } { ( n + 7 ) ! }$$

A) $\frac { n } { ( n + 7 ) ! }$
B) $\frac { 1 } { n + 7 }$
C) $\frac { n } { n + 7 }$
D) $\frac { n } { 7 }$

Question 7

Write out the first five terms of the sequence.
{2n}

A) 1, 4, 9, 16, 25
B) 2, 4, 8, 16, 32
C) 1, 2, 4, 8, 16
D) 4, 8, 16, 32, 64

Question 8

Evaluate the factorial expression.
$\frac { 7 ! } { 5 ! 2 ! }$

A) 0 !
B) 7
C) 21
D) 1

Question 9

Evaluate the factorial expression.
$\frac { 5 ! } { 3 ! }$

A) $2 !$
B) 20
C) 5
D) $\frac { 5 } { 3 }$

Question 10

The sequence is defined recursively. Write the first four terms.
$$a _ { 1 } = y ; a _ { n } = a _ { n - 1 } + U$$

A) $y , U , 2 U , 3 U$
B) $y , y - U , y - 2 U , y - 3 U$
C) $y , y + U , y + 2 U , y + 3 U$
D) $U , U + y , U + 2 y , U + 3 y$

Question 11

Evaluate the factorial expression.
$\frac { 8 ! } { 7 ! }$

A) 1
B) $\frac { 8 } { 7 }$
C) 8
D) $8 !$

Question 12

Write out the first five terms of the sequence.
{n2 - n}

A) 2, 6, 12, 20, 30
B) 1, 4, 9, 16, 25
C) 0, 3, 8, 15, 24
D) 0, 2, 6, 12, 20

Question 13

Evaluate the factorial expression.
$$\frac { 10 ! } { 5 ! 5 ! }$$

A) 504
B) 126
C) 30,240
D) 252

Question 14

Write out the first five terms of the sequence.
During a five-year period, a company doubles its profits each year. If the profits at the end of the fifth year are $ 224,000, then what are the profits for each of the first four years?

A) $14,000, $28,000, $42,000, $56,000
B) $14,000, $28,000, $56,000, $140,000
C) $14,000, $28,000, $56,000, $112,000
D) $15,000, $30,000, $60,000, $118,000

Question 15

The sequence is defined recursively. Write the first four terms.
a1 = 5 and a_n = 2a_n- 1 for n ≥ 2

A) 5, 9, 8, 7
B) 8, 16, 32, 64
C) 5, 12, 22, 42
D) 5, 10, 20, 40

Question 16

The sequence is defined recursively. Write the first four terms.
a1 = 135 and an+1 = 13 (an) for n ≥ 2

Question 17

The sequence is defined recursively. Write the first four terms.
$$a _ { 1 } = \sqrt { 6 } ; a _ { n } = \sqrt { 6 a _ { n - 1 } }$$

A) $\sqrt { 6 } , \sqrt { 6 \sqrt { 6 } } , \sqrt { 6 \sqrt { 6 \sqrt { 6 } } } , \sqrt { 6 \sqrt { 6 \sqrt { 6 \sqrt { 6 } } } }$
B) $\sqrt { 6 } , 6 \sqrt { 6 } , 36 \sqrt { 6 } , 216 \sqrt { 6 }$
C) $\sqrt { 6 } , 6,6 \sqrt { 6 } , 36$
D) $\sqrt { 6 } , \sqrt { \sqrt { 6 } } , \sqrt { \sqrt { \sqrt { 6 } } } , \sqrt { \sqrt { \sqrt { \sqrt { 6 } } } }$

Question 18

The sequence is defined recursively. Write the first four terms.
a1 = 2, a2 = 5 and a_n = a_n- 2 - 3a_n- 1 for n ≥ 3

A) 2, 5, -13, 44
B) 2, 5, -1, -16
C) 2, 5, 17, -46
D) 2, 5, 1, 2

Question 19

The sequence is defined recursively. Write the first four terms.
a1 = 5 and a_n = 4a_n- 1 - 1 for n ≥ 2

A) 5, 21, 85, 341
B) 5, 20, 80, 320
C) 5, 19, 75, 299
D) 5, 19, 79, 319

Question 20

The sequence is defined recursively. Write the first four terms.
a1 = 1 and a_n = a_n- 1 - 3 for n ≥ 2

A) -3, -6, -9, -12
B) 1, -2, -5, -8
C) 1, 4 , 7 , 10
D) 1, 0 , -3 , -6

Question 21

Write out the sum. Do not evaluate.
$$\sum _ { k = 1 } ^ { n } 7 ^ { k + 1 }$$

A) $7 + 7 ^ { 2 } + 7 ^ { 3 } + \ldots + 7 ^ { n + 1 }$
B) $7 + 7 ^ { 2 } + 7 ^ { 3 } + \ldots + 7 ^ { n }$
C) $7 ^ { 2 } + 7 ^ { 3 } + 7 ^ { 4 } + \ldots + 7 ^ { n }$
D) $7 ^ { 2 } + 7 ^ { 3 } + 7 ^ { 4 } + \ldots + 7 ^ { n + 1 }$

Question 22

Express the sum using summation notation.
$$\sum _ { k = 3 } ^ { 5 } \left( k ^ { 2 } - 8 \right)$$

A) 15
B) 0
C) 26
D) $- 12$

Question 23

Write out the sum. Do not evaluate.
$$\sum _ { k = 1 } ^ { 4 } \left( k ^ { 2 } - 4 k - 3 \right)$$

A) $- 6 - 7 - 6 - 3 + \ldots$
B) $- 6 - 7 - 6 - 3$
C) $- 6 + 7 - 6 + 3$
D) $1 - 6 - 7 - 6$

Question 24

Express the sum using summation notation.
$$\sum _ { k = 3 } ^ { 6 } 9 k$$

A) 81
B) 108
C) 54
D) 162

Question 25

Express the sum using summation notation.
$$\sum _ { k = 1 } ^ { 6 } ( - k )$$

A) $- 5$
B) $- 30$
C) $- 6$
D) $- 21$

Question 26

Express the sum using summation notation.
$$\sum _ { k = 2 } ^ { 4 } k ( k - 7 )$$

A) $- 3$
B) $- 40$
C) $- 22$
D) $- 34$

Question 27

Express the sum using summation notation.
$$\sum _ { k = 1 } ^ { 4 } \left( - \frac { 1 } { 4 } \right) ^ { k }$$

A) $= \frac { 47 } { 256 }$
B) $- \frac { 51 } { 256 }$
C) $\frac { 51 } { 256 }$
D) $\frac { 85 } { 256 }$

Question 28

Express the sum using summation notation.
$$2 + 8 + 18 + \ldots + 72$$

A) $\sum _ { k = 0 } ^ { 6 } 2 k ^ { 2 }$
B) $\sum _ { k = 1 } ^ { 6 } k ^ { 2 }$
C) $\sum _ { k = 1 } ^ { 6 } 2 ^ { 2 } k$
D) $\sum _ { k = 1 } ^ { 6 } 2 k ^ { 2 }$

Question 29

Express the sum using summation notation.
$$\sum _ { k = 1 } ^ { 4 } 2 ^ { k }$$

A) 14
B) 30
C) 20
D) 18

Question 30

Write out the sum. Do not evaluate.
$$\sum _ { k = 1 } ^ { 6 } ( 5 k - 2 )$$

A) 65
B) $3 + 8 + 13 + 18 + 23 + 28 + \ldots$
C) $1 + 3 + 8 + 13 + 18 + 23$
D) $3 + 8 + 13 + 18 + 23 + 28$

Question 31

Express the sum using summation notation.
$$\sum _ { k = 3 } ^ { 6 } ( 2 k - 5 )$$

A) 15
B) 13
C) 16
D) 21

Question 32

Write out the sum. Do not evaluate.
$$\sum _ { k = 1 } ^ { 4 } ( k + 4 )$$

A) $5 + 6 + 7 + 8 + \ldots$
B) $8 + 9 + 10 + 11$
C) $5 + 6 + 7 + 8$
D) $4 + 5 + 6 + 7$

Question 33

Write out the sum. Do not evaluate.
$$\sum _ { k = 1 } ^ { 5 } \frac { k + 1 } { k + 2 }$$

A) $\frac { 2 } { 3 } + \frac { 3 } { 4 } + \frac { 4 } { 5 } + \frac { 5 } { 6 } + \frac { 6 } { 7 }$
B) $\frac { 1 } { 3 } + \frac { 2 } { 4 } + \frac { 3 } { 5 } + \frac { 4 } { 6 }$
C) $\frac { 1 } { 2 } + \frac { 2 } { 3 } + \frac { 3 } { 4 } + \frac { 4 } { 5 } + \frac { 5 } { 6 }$
D) $\frac { 6 } { 7 }$

Question 34

Express the sum using summation notation.
$$\frac { 5 ^ { 14 } } { 7 ^ { 3 } } + \frac { 5 ^ { 13 } } { 7 ^ { 4 } } + \frac { 5 ^ { 12 } } { 7 ^ { 5 } } + \ldots + \frac { 5 ^ { 7 } } { 7 ^ { 10 } }$$

A) $\sum _ { k = 1 } ^ { 8 } \frac { 5 ^ { 15 - k } } { 7 ^ { 2 + k } }$
B) $\sum _ { k = 1 } ^ { 8 } \frac { 5 ^ { - k } } { 7 ^ { 2 + k } }$
C) $\sum _ { k = 1 } ^ { 4 } \frac { 5 ^ { 15 + k } } { 7 ^ { 2 + k } }$
D) $\sum _ { k = 1 } ^ { 8 } \frac { 5 ^ { 15 - k } } { 7 ^ { k } }$

Question 35

Express the sum using summation notation.
$$\frac { 1 } { 4 } + \frac { 2 } { 5 } + \frac { 1 } { 2 } + \cdots + \frac { 14 } { 17 }$$

A)
$$\sum _ { k = 0 } ^ { 14 } \frac { k } { k + 3 }$$
B)
$$\sum _ { k = 3 } ^ { 14 } \frac { k } { k + 1 }$$
C)
$$\sum _ { k = 1 } ^ { 14 } \frac { k } { k + 3 }$$
D)
$$\sum _ { k = 1 } ^ { n } \frac { k } { k + 3 }$$

Question 36

Express the sum using summation notation.
$$\sum _ { k = 1 } ^ { 5 } ( k - 10 )$$

A) $- 35$
B) $- 30$
C) $- 14$
D) $- 5$

Question 37

Express the sum using summation notation.
$$\sum _ { k = 1 } ^ { 16 } ( 2 k + 7 )$$

A) 53
B) 400
C) 384
D) 272

Question 38

Express the sum using summation notation.
$$3 ^ { 2 } + 4 ^ { 2 } + 5 ^ { 2 } + \cdots + 9 ^ { 2 }$$

A)
$\sum _ { k = 1 } ^ { 9 } k ^ { 2 }$
B)
$\sum _ { k = 4 } ^ { 9 } ( k - 1 ) ^ { 2 }$
C)
$\sum_{k=3}^{9} k^{2}$
D)
$\sum_{k=3}^{n} k^{2}$

Question 39

Express the sum using summation notation.
$$\frac { 4 } { 5 } - \frac { 16 } { 25 } + \frac { 64 } { 125 } - \cdots + ( - 1 ) ^ { ( 11 + 1 ) } \left( \frac { 4 } { 5 } \right) ^ { 11 }$$

A)
$$\sum _ { k = 1 } ^ { 11 } ( - 1 ) ^ { ( k + 1 ) } \left( \frac { 4 } { 5 } \right) ^ { k }$$
B)
$$\sum _ { k = 1 } ^ { 12 } ( - 1 ) ^ { ( k + 1 ) } \left( \frac { 4 } { 5 } \right) ^ { k }$$
C)
$\sum_{k=1}^{11}\left(\frac{4}{5}\right)^{k}$
D)
$\sum_{k=1}^{n}(-1)^{(k+1)}\left(\frac{4}{5}\right)^{k}$

Question 40

Express the sum using summation notation.
2 + 4 + 6 + ... + 16 A)
$\sum _ { k = 1 } ^ { 8 } k ^ { 2 }$
B)
$\sum _ { k = 0 } ^ { 8 } 2 k$
C)
$\sum _ { k = 1 } ^ { 8 } 2 k ^ { 2 }$
D)
$\sum _ { k = 1 } ^ { 8 } 2 k$

Question 41

Find the indicated term of the sequence.
The thirteenth term of the arithmetic sequence 28, 24, 20, ...

A) -24
B) -48
C) 76
D) -20

Question 42

Determine whether the sequence is arithmetic.
1, 4, 7, 10, 13, ...

A) Arithmetic
B) Not arithmetic

Question 43

Express the sum using summation notation.
$$\sum _ { k = 1 } ^ { 4 } \frac { 1 } { 7 k }$$

A) $\frac { 1 } { 28 }$
B) $\frac { 11 } { 42 }$
C) $\frac { 5 } { 28 }$
D) $\frac { 25 } { 84 }$

Question 44

Determine whether the sequence is arithmetic.
5, -15, 45, -135, 405, ...

A) Arithmetic
B) Not arithmetic

Question 45

Solve.
The number of students in a school in year n is estimated by the model a_n = 4n2 + 13n + 80. About how many students are in the school in each of the first three years?

A) 101, 122, 155
B) 110, 135, 168
C) 97, 122, 143
D) 97, 122, 155

Question 46

Determine whether the sequence is arithmetic.
4, 12, 36, 108, 972, ...

A) Arithmetic
B) Not arithmetic

Question 47

Find the indicated term of the sequence.
The twenty-first term of the arithmetic sequence 0, 14, 28, ...

A) 280
B) 260
C) 294
D) 308

Question 48

Express the sum using summation notation.
$$\sum _ { k = 3 } ^ { 6 } \frac { - 6 } { k }$$

A) 277
B) $- \frac { 57 } { 10 }$
C) $\frac { 17 } { 10 }$
D) $- \frac { 80 } { 27 }$

Question 49

An arithmetic sequence is given. Find the common difference and write out the first four terms.
{5n + 9}

A) d = 9; 5, 14, 19, 24
B) d = 5; 14, 19, 24, 29
C) d = 9; 14, 19, 24, 29
D) d = 5; 5, 14, 19, 24

Question 50

Find the indicated term of the sequence.
The twenty-fifth term of the arithmetic sequence 3 , -3 , -9 , ...

A) 147
B) -141
C) -147
D) 153

Question 51

Solve.
A town has a population of 20,000 people and is increasing by 10% every year. What will the population be at the end of 5 years?

A) 30,000 people
B) 32,210 people
C) 29,282 people
D) 12,210 people

Question 52

Express the sum using summation notation.
$$\sum _ { k = 1 } ^ { 4 } ( - 1 ) ^ { k } \cdot 2 k$$

A) $- 20$
B) 20
C) 16
D) 4

Question 53

Determine whether the sequence is arithmetic.
2, 4, 6, 10, 12, ...

A) Arithmetic
B) Not arithmetic

Question 54

Find the indicated term of the sequence.
The twentieth term of the arithmetic sequence -13, 23, 53, ...

A) -613
B) 593
C) 563
D) -583

Question 55

An arithmetic sequence is given. Find the common difference and write out the first four terms.
$\left\{ \frac { 1 } { 9 } + \frac { n } { 7 } \right\}$

A) $\mathrm { d } = \frac { 1 } { 9 } ; \frac { 1 } { 9 } , \frac { 16 } { 63 } , \frac { 25 } { 63 } , \frac { 34 } { 63 }$
B) $\mathrm { d } = \frac { 1 } { 7 } ; \frac { 1 } { 9 } , \frac { 16 } { 63 } , \frac { 25 } { 63 } , \frac { 34 } { 63 }$
C) $\mathrm { d } = \frac { 1 } { 7 } ; \frac { 16 } { 63 } , \frac { 25 } { 63 } , \frac { 34 } { 63 } , \frac { 43 } { 63 }$
D) $\mathrm { d } = \frac { 1 } { 9 } ; \frac { 16 } { 63 } , \frac { 25 } { 63 } , \frac { 34 } { 63 } , \frac { 43 } { 63 }$

Question 56

Express the sum using summation notation.
$$\sum _ { \mathrm { k } = 1 } ^ { 4 } ( - 1 ) ^ { \mathrm { k } } ( \mathrm { k } + 16 )$$

A) 74
B) $- 74$
C) 66
D) 2

Question 57

Determine whether the sequence is arithmetic.
2, -3, -8, -13, -18, ...

A) Arithmetic
B) Not arithmetic

Question 58

Determine whether the sequence is arithmetic.
2, -1, -4, -7, -10, ...

A) Arithmetic
B) Not arithmetic

Question 59

An arithmetic sequence is given. Find the common difference and write out the first four terms.
{12 - 6n}

A) d = -6; -6, -12, -18, -24
B) d = -6; 6, 2 , -4 , -10
C) d = -6; 6, 0, -6, -12
D) d = 6; 6, 12 , 18 , 24

Question 60

Express the sum using summation notation.
$$\sum _ { k = 2 } ^ { 5 } ( - 1 ) ^ { k + 1 } ( k - 1 ) ^ { 2 }$$

A) 10
B) 950
C) $- 30$
D) 30

Question 61

Find the sum of the arithmetic sequence.
2 + 4 + 6 + ... + 678

A) 115,260
B) 114,582
C) 115,600
D) 114,921

Question 62

Find the sum of the arithmetic sequence.
(-6) + (-1) + 4 + 9 + ... + 39

A) 330
B) 170
C) 160
D) 165

Question 63

Find the sum of the arithmetic sequence.
6 + 12 + 18 + ... + 492

A) 20172
B) 19926
C) 20418
D) 20667

Question 64

Find the sum of the arithmetic sequence.
1 + 2 + 3 + ... + 707

A) 250278
B) 500,556
C) 499,849
D) 249571

Question 65

Solve the problem.
The population of a town is increasing by 300 inhabitants each year. If its population at the beginning of 1990 was 23,626, what was its population at the beginning of 1997?

A) 330,638 inhabitants
B) 165,319 inhabitants
C) 25,726 inhabitants
D) 25,426 inhabitants

Question 66

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence.
6th term is -10; 15th term is -46

A) a1 = -10, d = -4, a_n = a_n- 1 - 4
B) a1 = 10, d = -4, a_n = a_n- 1 - 4
C) a1 = -30, d = 4, a_n = a_n- 1 + 4
D) a1 = 10, d = 4, a_n = a_n- 1 + 4

Question 67

Find the sum of the arithmetic sequence.
{3n + 7}, n = 48

A) 3864
B) 4032
C) 4200
D) 3792

Question 68

Find the indicated term using the given information.
a = -9 , d = -3 ; a8

A) 12
B) 15
C) -30
D) -33

Question 69

Find the indicated term using the given information.
-11 , -5 , 1 , ... ; a 27

A) 151
B) 145
C) -173
D) -167

Question 70

Find the indicated term using the given information.
a 17 = 34 , a11 = 16 ; a1

A) -8
B) 3
C) -14
D) -11

Question 71

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence.
7th term is -2; 15th term is -18

A) a1 = 12, d = -2, a_n = a_n- 1 - 2
B) a1 = 12, d = 2, a_n = a_n- 1 + 2
C) a1 = 10, d = 2, a_n = a_n- 1 + 2
D) a1 = 10, d = -2, a_n = a_n- 1 - 2

Question 72

Find the sum of the arithmetic sequence.
-3 + 1 + 5 + 9 + 13 + ... + (4n - 7)

A) n(2n + 5)
B) n(2n - 5)
C) n(4n - 7)
D) n(4n + 7)

Question 73

Find the indicated term using the given information.
$a _ { 14 } = - \frac { 17 } { 4 }$ , a $_ { 24 } = - \frac { 37 } { 4 } ; a _ { 4 }$

A) $- \frac { 1 } { 2 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 1 } { 4 }$
D) $- 9$

Question 74

Find the indicated term using the given information.
$$a = - 8 , d = \frac { 3 } { 4 } ; a_ {49}$$

A) 28
B) $- \frac { 179 } { 4 }$
C) $\frac { 115 } { 4 }$
D) $= 44$

Question 75

Find the sum of the arithmetic sequence.
1 + 3 + 5 + ... + 711

A) 126,025
B) 126,736
C) 127,449
D) 127,092

Question 76

Find the sum of the arithmetic sequence.
{5n - 1}, n = 49

A) 5953.5
B) 6272
C) 6443.5
D) 6076

Question 77

Find the indicated term using the given information.
a = 8 , d = 3 ; a34

A) -94
B) -91
C) 107
D) 110

Question 78

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence.
9th term is 74; 13th term is 114

A) a1 = -6, d = 10, a_n = a_n- 1 - 10
B) a1 = -16, d = 10, a_n = a_n- 1 + 10
C) a1 = -16, d = 10, a_n = a_n- 1 - 10
D) a1 = -6, d = 10, a_n = a_n- 1 + 10

Question 79

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence.
7th term is 29; 15th term is -27

A) a1 = 78, d = 7, a_n = a_n- 1 + 7
B) a1 = 71, d = -7, a_n = a_n- 1 - 7
C) a1 = 71, d = 7, a_n = a_n- 1 + 7
D) a1 = 78, d = -7, a_n = a_n- 1 - 7

Question 80

Find the indicated term using the given information.
a 13 = 80 , a16 = 98 ; a7

A) 6
B) 50
C) 44
D) 8