Deck 8: Sequences, Series, and Combinatorics

Question 1

Find the sum, if it exists.
$\sum _ { \mathrm { i } = 1 } ^ { 5 } 4 ( 3 ) ^ { \mathrm { i } }$

A) 2460
B) 93
C) 1452
D) Does not exist

Answer 1

C

Question 2

Use mathematical induction to prove the following.
$$1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$$

Answer 2

$$\begin{array} { l } 1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 \mathrm { k } - 2 ) ^ { 2 } = \frac { \mathrm { k } ^ { 2 } \left( 6 \mathrm { k } ^ { 2 } - 3 \mathrm { k } - 1 \right) } { 2 } \\1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 \mathrm { k } - 2 ) ^ { 2 } + [ 3 ( \mathrm { k } + 1 ) - 2 ] ^ { 2 } = \frac { \mathrm { k } \left( 6 \mathrm { k } ^ { 2 } - 3 \mathrm { k } - 1 \right) } { 2 } + [ 3 ( \mathrm { k } + 1 ) - 2 ] ^ { 2 } \\= \frac { 6 \mathrm { k } ^ { 3 } - 3 \mathrm { k } ^ { 2 } - \mathrm { k } } { 2 } + ( 3 \mathrm { k } + 1 ) ^ { 2 } \\= \frac { 6 \mathrm { k } ^ { 3 } - 3 \mathrm { k } ^ { 2 } - \mathrm { k } } { 2 } + \left( 9 \mathrm { k } ^ { 2 } + 6 \mathrm { k } + 1 \right) \\= \frac { 6 \mathrm { k } ^ { 3 } - 3 \mathrm { k } ^ { 2 } - \mathrm { k } } { 2 } + \frac { 18 \mathrm { k } 2 + 12 \mathrm { k } + 2 } { 2 } \\= \frac { 6 \mathrm { k } ^ { 3 } + 15 \mathrm { k } 2 + 11 \mathrm { k } + 2 } { 2 } \\= \frac { ( \mathrm { k } + 1 ) \left( 6 \mathrm { k } ^ { 2 } + 9 \mathrm { k } + 2 \right) } { 2 } \\= \frac { ( \mathrm { k } + 1 ) \left[ 6 ( \mathrm { k } + 1 ) ^ { 2 } - 3 ( \mathrm { k } + 1 ) - 1 \right] } { 2 } \\= \frac { ( \mathrm { k } + 1 ) \left[ 6 \left( \mathrm { k } ^ { 2 } + 2 \mathrm { k } + 1 \right) - ( 3 \mathrm { k } + 4 ) \right] } { 2 } \\= \frac { \left. \left( 6 \mathrm { k } ^ { 2 } + 12 \mathrm { k } + 6 \right) - ( 3 \mathrm { k } + 4 ) \right] } { 2 }\end{array}$$

Question 3

Find fraction notation.
$$0.8585 \overline { 85 }$$

A) $\frac { 17 } { 20 }$
B) $\frac { 85 } { 99 }$
C) $\frac { 17 } { 18 }$
D) $\frac { 81 } { 100 }$

Answer 3

B

Question 4

Use mathematical induction to prove the following.
$$\text { If } a \text { is } a \text { constant and } 0 < a < 1 \text {, then } a ^ { n } < a ^ { n - 1 } \text {. }$$

Question 5

Find the sum, if it exists.
$$\sum _ { i = 1 } ^ { \infty } 10 \left( - \frac { 5 } { 2 } \right) ^ { i - 1 }$$

A) $\frac { 2 } { 7 }$
B) $\frac { 20 } { 7 }$
C) $- \frac { 50 } { 7 }$
D) Does not exist

Question 6

Evaluate.
${ } _ { 42 } \mathrm { P } _ { 3 }$

A) 1722
B) $2,686,320$
C) 126
D) 68,880

Question 7

Use mathematical induction to prove the following.
$$1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \ldots + n ( n + 2 ) = \frac { n ( n + 1 ) ( 2 n + 7 ) } { 6 }$$

Question 8

Use mathematical induction to prove the following.
$$\sum _ { i = 1 } ^ { n } ( 5 i + 8 ) = \frac { n ( 5 n + 21 ) } { 2 }$$

Question 9

Use mathematical induction to prove the following.
$$\left( 3 ^ { 5 } \right) ^ { n } = 3 ^ { 5 } n$$

Question 10

Use mathematical induction to prove the following.
$$2 ^ { n } > n$$

Question 11

Use mathematical induction to prove the following.
If a is a constant and 0 < a < 1, then an < 1.

Question 12

Find the indicated term of the binomial expansion.
6th term; $\left( \frac { 4 } { m ^ { 3 } } + \frac { m ^ { 2 } } { 8 } \right) ^ { 8 }$

A) $\frac { 896 } { m ^ { 9 } }$
B) $\frac { 112 } { \mathrm {~m} ^ { 9 } }$
C) $\frac { 7 m ^ { 9 } } { 16 }$
D) $\frac { 56 m ^ { 9 } } { 16 }$

Question 13

Use mathematical induction to prove the following.
$$\left( 1 - \frac { 1 } { 2 } \right) \left( 1 - \frac { 1 } { 3 } \right) \ldots \left( 1 - \frac { 1 } { n + 1 } \right) = \frac { 1 } { n + 1 }$$

Question 14

Use mathematical induction to prove the following.
$$6 + 12 + 18 + \ldots + 6 n = 3 n ( n + 1 )$$

Question 15

Use mathematical induction to prove the following.
$$1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$$

Question 16

For the given arithmetic series, what is $\mathrm { S } _ { \mathrm { n } } ?$
$a _ { 1 } = 26 , d = 3$ , and $n = 30$

A) 8340
B) 2111
C) 4170
D) 2085

Question 17

Expand, where i $\AA ^ { 2 } = - 1 .$
$$( - 1 + i \sqrt { 3 } ) ^ { 3 }$$

A) 8
B) $- 1 - 3 \mathrm { i }$
C) $4 + 4 \mathrm { i }$
D) 1

Question 18

Write sigma notation.
$$4 + 5 + 6 + 7 + 8$$

A) $\sum _ { k = 0 } ^ { 5 } ( k + 3 )$
B) $\sum _ { k = 2 } ^ { 6 } ( k + 3 )$
C) $\sum _ { \mathrm { k } = 1 } ^ { 5 } ( \mathrm { k } + 3 )$
D) $\sum _ { k = 1 } ^ { 5 } k$

Question 19

Use mathematical induction to prove the following.
$$2 ^ { n } > 2 ^ { n - 1 }$$

Question 20

Use mathematical induction to prove the following.
$$0.64 n < 0.64 n - 1$$

Question 21

Find the first 4 terms of the recursively defined sequence.
$$a _ { 1 } = 2 , a _ { n } + 1 = \frac { ( - 1 ) ^ { n } } { a _ { n } }$$

A) $2 , - \frac { 1 } { 2 } , \frac { 1 } { 4 } , - \frac { 1 } { 8 }$
B) $- 2 , \frac { 1 } { 2 } , - \frac { 1 } { 3 } , \frac { 1 } { 4 }$
C) $2 , - \frac { 1 } { 2 } , - 2 , \frac { 1 } { 2 }$
D) $2 , - \frac { 1 } { 2 } , 2 , - \frac { 1 } { 2 }$

Question 22

Find the nth, or general, term.
$$\frac { 1 } { 7 } , \frac { 1 } { 49 } , \frac { 1 } { 343 } , \ldots$$

A) $7 ^ { n - 1 }$
B) $\frac { 1 } { 7 ^ { n - 1 } + 6 }$
C) $\frac { 1 } { 7 + 42 ( n - 1 ) }$
D) $\frac { 1 } { 7 ^ { n } }$

Question 23

Find the sum.
$$\sum _ { i = 1 } ^ { 6 } ( 5 i - 4 )$$

A) 81
B) 26
C) 6
D) 75

Question 24

Evaluate the sum.
$\sum _ { i = 0 } ^ { 5 } 3 ^ { i }$

A) 202
B) 364
C) 355
D) 363

Question 25

Find the nth, or general, term.
$\frac { 1 } { 7 } , \frac { 9 } { 7 } , \frac { 81 } { 7 } , \ldots$

A) $\frac { 1 } { 7 } + 9 ( \mathrm { n } - 1 )$
B) $\frac { 1 } { 7 } ( 9 ) ^ { n - 1 }$
C) $\frac { 1 } { 7 } + \frac { 8 } { 7 } ( \mathrm { n } - 1 )$
D) $\left( \frac { 1 } { 7 } \right) ^ { n - 1 } + \frac { 8 } { 7 }$

Question 26

Solve.
How many 2-letter codes can be formed with the letters P, Q, R, S, T without repetition?

A)20
B)4
C)10
D)40

Question 27

What term of the arithmetic sequence is the given number?
3.42, 3.57, 3.72, . . . ;5.97

A)17
B)16
C)15
D)18

Question 28

Find the indicated term of the sequence.
$$a _ { n } = \frac { n } { n ^ { 2 } + 2 } ; a 90$$

A) $\frac { 90 } { 4051 }$
B) $\frac { 5 } { 2026 }$
C) $\frac { 45 } { 4051 }$
D) $\frac { 45 } { 8102 }$

Question 29

Find the indicated sum.
Find the sum of the first 13 terms of the geometric sequence: $\frac { 1 } { 2 } , - 1,2 , - 4,8 , \ldots$

A) $\frac { 2729 } { 2 }$
B) 1369
C) $\frac { 2725 } { 2 }$
D) $\frac { 2731 } { 2 }$

Question 30

Evaluate.
$\frac { 4 ! } { 2 ! 2 ! }$

A) 0 !
B) 4
C) 1
D) 6

Question 31

Solve.
Find the sum of the first 114 positive multiples of 3.

A)19,323
B)19,665
C)19,494
D)19,838

Question 32

Solve the problem.
The population of a town was 25,400 at the beginning of 1970. If the population decreased 350 people per year, how many people lived in the town at the beginning of 1985?

A)19,800 people
B)20,150 people
C)5250 people
D)20,500 people

Question 33

Find the sum, if it exists.
$$0.1 + 0.01 + 0.001 + 0.0001 + \ldots$$

A) $\frac { 111 } { 1000 }$
B) $\frac { 1 } { 9 }$
C) $\frac { 11 } { 100 }$
D) Does not exist

Question 34

Expand, where i $i ^ { 2 } = - 1$
$$\left( \frac { 1 } { \sqrt { 2 } } + \frac { i } { \sqrt { 2 } } \right) 4$$

A) $1 - \mathrm { i }$
B) $- 1$
C) 1
D) $1 + \mathrm { i }$

Question 35

Determine the number of subsets of the set.
A set of 11 elements

A)2048
B)22
C)1024
D)23

Question 36

Write sigma notation.
$$0 + 2 + 6 + 12 + 20$$

A) $\sum _ { i = 1 } ^ { 5 } ( 2 i - 2 )$
B) $\sum _ { i = 1 } ^ { 5 } \left( i ^ { 2 } - i \right)$
C) $\sum _ { i = 1 } ^ { 5 } ( 2 i + 2 )$
D) $\sum _ { i = 1 } ^ { 5 } \left( i ^ { 2 } + i \right)$

Question 37

Evaluate.
${ } _ { 7 } \mathrm { P } _ { 0 }$

A) $0.5$
B) 10,080
C) 5040
D) 1

Question 38

Find the common ratio.
64, 32, 16, 8, 4, . . .

A)4
B)-0.5
C)2
D)0.5

Question 39

Find the indicated term.
$4,4 \sqrt { 2 } , 8 , \ldots$ ; the 9 th term.

A) $64 \sqrt { 2 }$
B) $32 \sqrt { 2 }$
C) 64
D) 128

Question 40

Find fraction notation.
$0.0 \overline { 1 }$

A) 0
B) $\frac { 1 } { 50 }$
C) $54 \frac { 1 } { 9 }$
D) $\frac { 1 } { 90 }$

Question 41

Solve.
A sequence of yearly payments of $3000 is invested at the end of each of 9 years at an interest rate of 4.6%%, compounded annually. What is the total amount of the annuity after 9 years?

A)$32,539.77
B)$31,539.77
C)$33,539.77
D)$32,039.77

Question 42

Find the indicated quantity.
$a _ { 1 }$ , when $d = - 3$ and $a _ { 36 } = - 93$

A) $- 3$
B) 9
C) 6
D) 12

Question 43

Find the nth, or general, term.
$25,5,1 , \ldots$

A) $5 ^ { n - 1 }$
B) $5 ^ { n }$
C) $5 ^ { 3 - n }$
D) $5 ( \mathrm { n } - 1 )$

Question 44

Solve the problem.
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, $112,000
C)$15,000, $30,000, $60,000, $118,000
D)$14,000, $28,000, $56,000, $140,000

Question 45

Evaluate the sum.
$$\sum _ { \mathrm { k } = 0 } ^ { 10 } ( - 1 ) ^ { \mathrm { k } }$$

A) 1
B) 10
C) 11
D) 0

Question 46

Evaluate.
$\left( \begin{array} { c } 12 \\ 5 \end{array} \right)$

A) 330
B) 3960
C) 792
D) 66

Question 47

Solve the problem.
An experiment was conducted. The results are listed in the following table. $$\begin{array} { c | r } \text { Event } & \text { Number of occurrences } \\\hline \text { A } & 64 \\\text { B } & 110 \\\text { C } & 93 \\\text { D } & 85 \\\text { E } & 148\end{array}$$ Use the table to determine which event has the greatest probability of occurring and what the probability of its occurrence is.

A)E; 0.296
B)D; 0.27
C)A; 0.228
D)D; 0.17

Question 48

Solve the problem.
$46 ^ { \mathrm { P } } 10$

A) $1.479 \times 10 ^ { 16 }$
B) $1.479 \times 10 ^ { 17 }$
C) $2.958 \times 10 ^ { 16 }$
D) $1.479 \times 10 ^ { 15 }$

Question 49

Find the indicated term of the sequence.
$$a _ { n } = ( 5 n - 7 ) ( 6 n - 5 ) ; a _ { 12 }$$

A) 5159
B) 3551
C) 4081
D) 4489

Question 50

Determine how many of the first 5 statements in the sequence obtainable from the given statement are true.
The measure of each interior angle of a regular polygon with $n$ sides is $180 - \frac { 360 } { n }$ . $( n \geq 3 )$

A) 4
B) 5
C) 2
D) 3

Question 51

Solve.
How many 4-letter codes can be formed with the letters A, B, C, D, E, F, G, H with repetition?

A)4096
B)336
C)240
D)56

Question 52

The nth term of a sequence is given. Find the first 4 terms.
$$a _ { n } = 8 n - 1$$

A) $8,15,23,31$
B) $9,15,23,31$
C) $7,15,23,31$
D) $7,17,23,33$

Question 53

Find the common ratio.
$$\frac { 3 } { 2 } , \frac { 3 } { 8 } , \frac { 3 } { 32 } , \frac { 3 } { 128 } , \frac { 3 } { 512 } , \ldots$$

A) 4
B) 20
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 20 }$

Question 54

Find the first 4 terms of the recursively defined sequence.
$$a _ { 1 } = 4 , a _ { n } + 1 = \left( a _ { n } \right) ^ { 2 }$$

A) $4,8,16,32$
B) $4,16,64,256$
C) $4,4,9,16$
D) $4,16,256,65,536$

Question 55

For the given arithmetic series, what is $\mathrm { S } _ { \mathrm { n } } ?$
$a _ { 1 } = 21 , d = - 3$ , and $n = 17$

A) $- 204$
В) $- 51$
C) $- 102$
D) $- 30$

Question 56

Find the indicated term of the binomial expansion.
7 th term; $( x + 2 y ) ^ { 10 }$

A) $6720 x ^ { 6 } y ^ { 4 }$
B) $13,440 x ^ { 4 } y ^ { 6 }$
C) $13,440 x ^ { 6 } y ^ { 4 }$
D) $6720 x ^ { 4 } y^7$

Question 57

Evaluate the sum.
$$\sum _ { \mathrm { k } = 1 } ^ { 4 } ( - 1 ) ^ { \mathrm { k } } ( \mathrm { k } + 3 )$$

A) 22
B) 14
C) 2
D) $- 22$

Question 58

Solve the problem.
$20 C _ { 4 }$

A) 19,380
B) 4845
C) 29,070
D) 116,280

Question 59

Solve.
Suppose 6 cards are drawn from a deck of 52 cards. What is the probability of drawing 3 spades and 3 hearts? A) $\frac { 1296 } { 72,709 }$
В) $\frac { 1573 } { 391,510 }$
C) $\frac { 3 } { 5,089,630 }$
D) $\frac { 1573 } { 49,980 }$

Question 60

Expand.
$$\left( \frac { 5 } { x } - 3 x \right) ^ { 4 }$$

A) $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 2 } } + 1350 x - 540 x ^ { 2 } + 81 x ^ { 4 }$
B) $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 2 } } + 2700 - 540 x ^ { 2 } + 81 x ^ { 4 }$
C) $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 3 } } + 1350 - 540 x ^ { 3 } + 81 x ^ { 4 }$
D) $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 2 } } + 1350 - 540 x ^ { 2 } + 81 x ^ { 4 }$

Question 61

Expand.
$( 1 - \sqrt { 3 } ) ^ { 4 } - ( 1 + \sqrt { 3 } ) ^ { 4 }$

A) $- 7 \sqrt { 3 }$
B) $- 55$
C) $- 32 \sqrt { 3 }$
D) $- 28$

Question 62

Find the sum, if it exists.
$$8 - 9 + \frac { 81 } { 8 } - \frac { 729 } { 64 } + \ldots$$

A) $- \frac { 64 } { 17 }$
B) $\frac { 72 } { 17 }$
C) $- \frac { 72 } { 17 }$
D) Does not exist

Question 63

Evaluate.
$\frac { 5 ! } { 3 ! }$

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

Question 64

Evaluate.
$\left( \begin{array} { c } 15 \\ 1 \end{array} \right)$

A) 210
B) 30
C) $1.308 \times 10 ^ { 12 }$
D) 15

Question 65

Find the first term and the common difference.
$6,9,12,15 , \ldots$

A) $a _ { 1 } = 6 , d = 15$
B) $a _ { 1 } = 6 , d = 3$
C) $a _ { 1 } = 6 , d = 9$
D) $a _ { 1 } = 6 , d = - 3$

Question 66

Find the first term and the common difference.
$$\frac { 7 } { 12 } , \frac { 1 } { 3 } , \frac { 1 } { 12 } , - \frac { 1 } { 6 } , \ldots$$

A) $\mathrm { a } _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = \frac { 1 } { 4 }$
B) $\mathrm { a } _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = - \frac { 1 } { 4 }$
C) $a _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = \frac { 7 } { 12 }$
D) $\mathrm { a } _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = - \frac { 3 } { 4 }$

Question 67

Solve.
A pendulum bob swings $5.0 \mathrm {~cm}$ on its first oscillation. On each subsequent oscillation the bob travels $\frac { 2 } { 5 }$ of the previous distance. Find the total distance the bob travels before coming to rest.

A) $12.4 \mathrm {~cm}$
B) $4.1 \mathrm {~cm}$
C) $11.7 \mathrm {~cm}$
D) $3.3 \mathrm {~cm}$

Question 68

Expand.
$$( \sqrt { 3 } + 1 ) ^ { 6 } + ( \sqrt { 3 } - 1 ) ^ { 6 }$$

A) $370 \sqrt { 3 }$
B) 370
C) 208
D) 416

Question 69

Evaluate.
${ } _ { 20 } \mathrm { P } _ { 4 }$

A) 6840
B) $2.43 \times 10^{18}$
C) 116,280
D) $1,860,480$

Question 70

What term of the arithmetic sequence is the given number?
$$60.04,75.03,90.02 , \ldots ; 329.86$$

A) 22
B) 21
C) 19
D) 20

Question 71

Find the indicated term of the sequence.
$$a _ { n } = 5 n ^ { 2 } ( 15 n - 118 ) ; a _ { 6 }$$

A) $- 6120$
B) $- 5040$
C) 38,520
D) 37,440

Question 72

Find the nth, or general, term.
1, 7, 49, . . . A) $7 ^ { n - 1 }$
В) $7 ^ { 1 - n }$
C) $7 ^ { \mathrm { n } }$
D) $7 ( \mathrm { n } - 1 )$

Question 73

Solve the problem.
An experiment was conducted. The results are listed in the following table.



What are the values of c and f?

A)0.144, 0.16
B)0.16, 0.21
C)0.14, 0.16
D)0.14, 0.26

Question 74

Find the indicated term of the binomial expansion.
5 th term; $( 2 x + 5 ) ^ { 5 }$

A) $2500 x ^ { 2 }$
B) $6250 x$
C) 15,625
D) $1250 x$

Question 75

Find the indicated quantity.
$a _ { 31 }$ , when $\mathrm { a } _ { 1 } = 1$ and $\mathrm { d } = - \frac { 7 } { 6 }$

A) $\frac { 223 } { 6 }$
B) 36
C) $- 34$
D) $- \frac { 211 } { 6 }$

Question 76

Determine how many of the first 5 statements in the sequence obtainable from the given statement are true.
$n ^ { 2 } + 2 n + 1$ is a perfect square.

A) 4
B) 2
C) 3
D) 5

Question 77

Find fraction notation.
$0.888 \overline { 8 }$

A) $\frac { 22 } { 25 }$
B) $\frac { 8 } { 9 }$
C) $\frac { 111 } { 125 }$
D) $\frac { 9 } { 10 }$

Question 78

Determine how many of the first 5 statements in the sequence obtainable from the given statement are true.
The measure of each exterior angle of a regular polygon with $n$ sides is $\frac { 360 } { n }$ . ( $\left. n \geq 3 \right)$

A) 5
B) 3
C) 2
D) 4

Question 79

Expand.
$$( x - \sqrt { 7 } ) ^ { 5 }$$

A) $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 70 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 49 x - 49 \sqrt { 7 }$
B) $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 70 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 245 x - 49$
C) $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 70 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 245 x - 49 \sqrt { 7 }$
D) $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 7 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 245 x - 49 \sqrt { 7 }$

Question 80

Solve.
In how many distinguishable ways can the letters of the word WINDOW be arranged?

A)120
B)720
C)360
D)36