Deck 11: Sequences, Series, and the Binomial Theorem

Question 1

Find the indicated term for the sequence.

-Find the $5^{\text {th }}$ term of the sequence whose general term is $a_{n}=3(2 n-2)$ .

A) 18
B) 24
C) 40
D) 30

Answer 1

24

Question 2

Find the indicated term for the sequence.

-Find the $12^{\text {th }}$ term of the sequence whose general term is $a_{n}=3 n-1$ .

A) 36
B) 35
C) 37
D) 24

Answer 2

35

Question 3

Find the indicated term for the sequence.

-Find the $11^{\text {th }}$ term of the sequence whose general term is $a_{n}=n^{2}-n$ .

A) 11
B) 132
C) 110
D) -110

Answer 3

110

Question 4

Find the indicated term for the sequence.

-Find the $5^{\text {th }}$ term of the sequence whose general term is $a_{n}=4^{n}$ .

A) 1024
B) 625
C) 256
D) 20

Question 5

Find the indicated term for the sequence.

-Find the $12^{\text {th }}$ term of the sequence whose general term is $a_{n}=(3 n-8)(5 n+5)$ .

A) 1500
B) 1820
C) 2170
D) 1210

Question 6

Find the indicated term for the sequence.

-Find the $12^{\text {th }}$ term of the sequence whose general term is $a_{n}=\frac{2 n-1}{3 n+5}$ .

A) $\frac{23}{38}$

B) $\frac{23}{41}$

C) $\frac{21}{41}$

D) $\frac{21}{38}$

Question 7

Find the first five terms of the sequence with the given general term.

- $a_{n}=n-3$

A) $2,1,0,-1,-2$
B) $-2,-1,0,1,2$
C) $-3,-2,-1,0,1$
D) $2,1,0,1,2$

Question 8

Find the first five terms of the sequence with the given general term.

- $a_{n}=(-1)^{n+1}$

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

Question 9

Find the general term $a_n$ of the given sequence

- $2,6,10,14,18, \ldots$

A) $a_{n}=2 n-4$
B) $\mathrm{a}_{\mathrm{n}}=\mathrm{n}+4$
C) $a_{n}=4 n-2$
D) $a_{n}=2(4)^{n-1}$

Question 10

Find the general term $a_n$ of the given sequence

- $3,15,27,39,51, \ldots$

A) $a_{n}=9 n-12$
B) $a_{n}=12 n-3$
C) $a_{n}=3(4 n-3)$
D) $a_{n}=3(12)^{n-1}$

Question 11

Find the general term $a_n$ of the given sequence

- $3,9,27,81,243, \ldots$

A) $a_{n}=3+6(n-1)$
B) $a_{n}=6 n$
C) $a_{n}=3^{n}$
D) $a_{n}=3^{n-1}+2$

Question 12

Find the general term $a_n$ of the given sequence

- $4,-1, \frac{1}{4},-\frac{1}{16}, \frac{1}{64}$ .

A) $a_{n}=4\left(\frac{1}{4}\right)^{n-1}$

B) $a_{n}=\frac{1}{4}(-4) n-1$

C) $a_{n}=4\left(-\frac{1}{4}\right)^{n-1}$

D) $a_{n}=6\left(-\frac{1}{6}\right)^{n-1}$

Question 13

Examine the differences for the sequence, and determine the type of expression

- $2,2,8,18,32$

A) quadratic
B) cubic
C) linear

Question 14

Examine the differences for the sequence, and determine the type of expression

- $8,5,2,-1,-4$

A) quadratic
B) linear
C) cubic

Question 15

Examine the differences for the sequence, and determine the type of expression

- $1,0,-1,-8,-27$

A) cubic
B) linear
C) quadratic

Question 16

Find the first four terms of the given sequence.

- $a_{1}=3, a_{n}=a_{n}-1+8$

A) $11,19,27,35$
B) 3, 11, 19, 27
C) $3,8,16,24$
D) $0,8,16,24$

Question 17

Find the first four terms of the given sequence.

- $\mathrm{a}_{1}=9, \mathrm{a}_{\mathrm{n}}=2 \cdot \mathrm{a}_{\mathrm{n}}-1$

A) $18,36,72,144$
B) $0,4,36,40$
C) $9,36,40,44$
D) $9,18,36,72$

Question 18

Find the first four terms of the given sequence.

- $a_{1}=5, a_{n}=\frac{(-1)^{n-1}}{a_{n}-1}$

A) - $5, \frac{1}{2},-\frac{1}{3}, \frac{1}{4}$

B) $5,-\frac{1}{5},-5, \frac{1}{5}$

C) $5,-\frac{1}{5}, 5,-\frac{1}{5}$

D) $5,-\frac{1}{5}, \frac{1}{25},-\frac{1}{125}$

Question 19

Find the first four terms of the given sequence.

- $a_{1}=3, a_{n}=\left(a_{n}-1\right)^{2}$

A) $3,9,81,6561$
B) $3,6,12,24$
C) $3,9,27,81$
D) $3,4,9,16$

Question 20

The Fibonacci sequence is a sequence whose first two terms are $f_{1}=1$ and $f_{2}=1$ , and all other terms are the sum of the two preceding terms. The terms of the Fibonacci sequence are often referred to as Fibonacci numbers.
-One interesting fact about the Fibonacci numbers is that for any $n \geq 3$ ,
$2 \cdot f_{n}-f_{n+1}=f_{n}-2$ . Select a Fibonacci number, and verify this fact.

Question 21

The Fibonacci sequence is a sequence whose first two terms are $f_{1}=1$ and $f_{2}=1$ , and all other terms are the sum of the two preceding terms. The terms of the Fibonacci sequence are often referred to as Fibonacci numbers.
-The sum of any 10 consecutive Fibonacci numbers is divisible by 11 . Select 10 Fibonacci numbers, and verify this fact.

Question 22

Find the indicated partial sum for the given sequence

- $s_{5}, 4,2,0,-2, \ldots$

A) -6
B) 4
C) 20
D) 0

Question 23

Find the indicated partial sum for the given sequence

- $\mathrm{s} _5,-15,-18,-21,-24, \ldots$

A) -15
B) -30
C) -105
D) -75

Question 24

Find the indicated partial sum for the given sequence

-s10, 86, 84, 82, 80, ..

A) 770
B) 340
C) -340
D) 950

Question 25

Find the indicated partial sum for the given sequence

- $\mathrm{s} _{10},-17,54,125, \ldots$

A) -3110
B) -3365
C) 3110
D) 3025

Question 26

Find the indicated partial sum for the given sequence

-s6, 729, 2187, 6561, 19,683, ..

A) 265356
B) 796797
C) $-\frac{88209}{2}$
D) 177,147

Question 27

Find the indicated partial sum for the given sequence

- $\mathrm{s}_{5},-\frac{1}{8}, \frac{1}{16},-\frac{1}{32}, \frac{1}{64}, \ldots$

A) $-\frac{1}{128}$

B) $-\frac{21}{256}$

C) $-\frac{15}{256}$

D) $-\frac{11}{128}$

Question 28

Find the indicated partial sum for the given sequence

- $s_{6}, 3,-8,13,-18, \ldots$

A) -12
B) - 15
C) 15
D) 8

Question 29

Find the indicated partial sum for the given sequence

- $s_{6}, 2,-2,2,-2, \ldots$

A) 2
B) 4
C) -2
D) 0

Question 30

Find the indicated partial sum for the sequence with the given general term

- $s_{4}, a_{n}=-3 n+7$

A) 16
B) 4
C) -2
D) -5

Question 31

Find the indicated partial sum for the sequence with the given general term

- $\mathrm{s} 4, \mathrm{a}_{\mathrm{n}}=2 \mathrm{n}-9$

A) -16
B) -1
C) -7
D) -28

Question 32

Find the indicated partial sum for the sequence with the given general term

- $\mathrm{s}_{5}, \mathrm{a}_{\mathrm{n}}=-4 \mathrm{n}+9$

A) 5
B) -15
C) -11
D) 25

Question 33

Find the indicated partial sum for the sequence with the given general term

- $\mathrm{s}_{6}, \mathrm{a}_{\mathrm{n}}=-5 \mathrm{n}-8$

A) -13
B) -153
C) -78
D) - 38

Question 34

Find the indicated partial sum for the sequence with the given general term

- $s_{4}, a_{n}=(-1)^{n}$

A) 1
B) 0
C) -4
D) -1

Question 35

Find the indicated partial sum for the sequence with the given general term

- $\mathrm{s}_{3}, \mathrm{a}_{\mathrm{n}}=(\mathrm{n}+2)(\mathrm{n}-6)$

A) 46
B) -46
C) 98
D) -31

Question 36

Find the indicated partial sum for the given sequence with the given first term and general term

- $\mathrm{s}_{4}, \mathrm{a}_{1}=4, \mathrm{a}_{\mathrm{n}}=\mathrm{a}_{\mathrm{n}}-1+7$

A) 90
B) 58
C) 33
D) 54

Question 37

Find the indicated partial sum for the given sequence with the given first term and general term

- $\mathrm{s}_{5}, \mathrm{a}_{1}=-1, \mathrm{a}_{\mathrm{n}}=-10 \mathrm{a}_{\mathrm{n}}-1$

A) - 950
B) 1049
C) 1050
D) - 9091

Question 38

Find the indicated partial sum for the given sequence with the given first term and general term

- $\mathrm{s}_{4}, \mathrm{a}_{1}=1, \mathrm{a}_{\mathrm{n}}=\left(\mathrm{a}_{\mathrm{n}}-1\right)^{2}+2$

A) 10
B) 136
C) 6
D) 138

Question 39

Find the sum

- $\sum_{i=2}^{5}(4 i-4)$

A) 36
B) 24
C) 40
D) 32

Question 40

Find the sum

- $\sum_{i=1}^{6}(6 i-3)$

A) 76
B) 75
C) 108
D) 1165

Question 41

Find the sum

- $\sum_{i=3}^{6}\left(i^{2}-3\right)$

A) 86
B) 41
C) 74
D) 99

Question 42

Find the sum

- $\sum_{i=3}^{6} 3^{i}$

A) 1080
B) 729
C) 162
D) 19,683

Question 43

Find the sum

- $\sum_{\mathrm{i}=3}^{6} \frac{\left(\mathrm{i}^{2}-3\right)}{2}$

A) 20
B) 37
C) 74
D) 39

Question 44

Find the sum

- $\sum_{i=1}^{4} \frac{i+1}{i+2}$

A) $\frac{6}{7}$

B) $\frac{61}{20}$

C) $\frac{71}{20}$

D) $\frac{21}{10}$

Question 45

Find the sum

- $\sum_{i=1}^{4}\left(i^{2}-5 i-3\right)$

A) 0
B) -32
C) -24
D) 13

Question 46

Solve.

-One approximation for $\pi$ is the infinite sum $\sum_{i=1}^{\infty}\left[(-1)^{i}+1 \cdot \frac{4}{2 i-1}\right]$ . Find the sum of the first 9 terms, rounded to the nearest hundredth.

A) 3.22
B) 3.28
C) 3.25
D) 3.24

Question 47

Solve.

-One approximation for Euler's number e is the infinite sum $1+\sum_{\mathrm{i}=1}^{\infty} \frac{1}{\mathrm{i} !}=1+$ $\left.\left(\frac{1}{1}+\frac{1}{1 \cdot 2}+\frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{1 \cdot 2 \cdot 3 \cdot 4}+\ldots\right).\right)$ The expression $\mathrm{i}$ ! is called $\mathrm{i}$ factorial and is equal to the product of the integers from 1 through $i$ . Find the sum through $i=9$ , rounded to the nearest thousandth.

A) 2.717
B) 2.718
C) 2.708
D) 2.709

Question 48

Write the sum using summation notation

- $0+1+2+3+4$

A) $\sum_{i=1}^{5}(i-1)$

B) $\sum_{i=1}^{4}(i-1)$

C) $\sum_{i=1}^{5}(n-1)$

D) $\sum_{i=1}^{5} \mathrm{i}$

Question 49

Write the sum using summation notation

- $7+12+17+22+27$

A) $\sum_{i=1}^{5}(i+2)$

B) $\sum_{i=1}^{5} 5 \mathrm{i}$

C) $\sum_{i=1}^{4}(5 i+2)$

D) $\sum_{i=1}^{5}(5 i+2)$

Question 50

Write the sum using summation notation

- $0+2+6+12+20$

A) $\sum_{i=1}^{5}\left(i^{2}+\mathrm{i}\right)$

B) $\sum_{i=1}^{5}\left(i^{2}+i-2\right)$

C) $\sum_{i=1}^{5}\left(i^{2}-i\right)$

D) $\sum_{i=1}^{5}(2 i-2)$

Question 51

Write the sum using summation notation

- $18+30+42+54+66$

A) $\sum_{i=1}^{5}(12 i+6)$

B) $\sum_{i=1}^{5}(12 n+6)$

C) $\sum_{i=1}^{6}(12 i+6)$

D) $\sum_{i=0}^{5}(12 i+6)$

Question 52

Write the sum using summation notation

- $\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}$

A) $\sum_{i=0}^{4} \frac{1}{i+5}$

B) $\sum_{\mathrm{i}=1}^{5} \frac{1}{\mathrm{i}+6}$

C) $\sum_{i=0}^{4} \frac{1}{i+6}$

D) $\sum_{i=1}^{5} \frac{1}{i+5}$

Question 53

Solve the problem.

-The number of students in a school in year $n$ is estimated by the model $a_{n}=6 n^{2}+13 n+83$ . Write a sequence showing how many students are in the school in each of the first three years.

A) $102,133,176$
B) $108,133,176$
C) $115,146,189$
D) $102,133,158$

Question 54

Solve the problem.

-A man earned $\$ 2500$ the first year he worked. If he received a raise of $\$ 400$ at the end of each year, what was his salary during the 15th year?

A) $\$ 5600$
B) $\$ 8100$
C) $\$ 8500$
D) None of these

Question 55

Solve the problem.

-The population of a town was 35,200 at the beginning of 1970 . If the population decreased 300 people per year, how many people lived in the town at the beginning of 1985 ?

A) 30,400
B) 4500
C) 30,700
D) 31,000

Question 56

Solve the problem.

-A town has a population of 1000 people and is increasing by $9 \%$ every year. What will the population be at the end of 11 years?

A) 2813
B) 1990
C) 2367
D) 2580

Question 57

Find the common difference,d, of the given arithmetic sequence

- $4,6,8,10, \ldots$

A) 4
B) 1.5
C) 2

Question 58

Find the common difference,d, of the given arithmetic sequence

- $5,9,13,17, \ldots$

A) 4
B) -4
C) 12

Question 59

Find the common difference,d, of the given arithmetic sequence

- $-8,-13,-18,-23, \ldots$

A) -10
B) -15
C) -5

Question 60

Find the common difference,d, of the given arithmetic sequence

- $\frac{1}{2},-\frac{1}{2},-\frac{3}{2},-\frac{5}{2}, \ldots$

A) 5
B) - 1
C) -2

Question 61

Find the first five terms of the arithmetic sequence with the given first term and common difference

- $\mathrm{a}_{1}=17, \mathrm{~d}=4$

A) $17,4,25,29,33$
B) $4,21,38,55,72$
C) $17,21,25,29,33$
D) $17,4,42,63,84$

Question 62

Find the first five terms of the arithmetic sequence with the given first term and common difference

- $\mathrm{a}_{1}=12, \mathrm{~d}=-2$

A) $16,13,10,7,4$
B) $12,10,8,6,4$
C) $0,12,10,8,6$
D) $-12,-10,-8,-6,-4$

Question 63

Find the first five terms of the arithmetic sequence with the given first term and common difference

- $\mathrm{a}_{1}=6, \mathrm{~d}=-1$

A) $5,4,3,2,1$
B) $6,5,4,3,2$
C) $6,5,3,3,2$
D) $7,6,5,4,3$

Question 64

Find the first five terms of the arithmetic sequence with the given first term and common difference

- $\mathrm{a}_{1}=-17, \mathrm{~d}=4$

A) $-1,-5,-9,-13,-17$
B) - $17,-13,-9,-5,-1$
C) $-9,-5,-1,3,7$
D) $-9,-13,-17,-21,-25$

Question 65

Find the general term, $a_n$ , of the given arithmetic sequence

- $2,4,6,8,10, \ldots$

A) $a_{n}=2 n$
B) $a_{n}=3 n$
C) $a_{n}=n^{2}$
D) $a_{n}=n+2$

Question 66

Find the general term, $a_n$ , of the given arithmetic sequence

- $2,6,10,14,18, \ldots$

A) $a_{n}=2 n-4$
B) $a_{n}=n+4$
C) $a_{n}=4 n-2$
D) $a_{n}=2(4)^{n-1}$

Question 67

Find the general term, $a_n$ , of the given arithmetic sequence

- $3,12,21,30,39, \ldots$

A) $a_{n}=3(3 n-2)$
B) $a_{n}=6 n-9$
C) $a_{n}=3(9)^{n-1}$
D) $a_{n}=9 n-2$

Question 68

Find the general term, $a_n$ , of the given arithmetic sequence

- $\frac{7}{8}, \frac{8}{9}, \frac{9}{10}, \frac{10}{11}, \frac{11}{12}, \ldots$

A) $a_{n}=\frac{n+6}{n-7}$

B) $a_{n}=\frac{6}{n+7}$

C) $a_{n}=\frac{n+6}{n+7}$

D) $a_{n}=\frac{n+7}{n+6}$

Question 69

Find the indicated term.

-Find the 17th term of the arithmetic sequence 5, 8, 11, ....

A) -43
B) 48
C) 56
D) 53

Question 70

Find the indicated term.

-Find the 15th term of the arithmetic sequenœe - 4, - 7, - 10, ...

A) -46
B) -42
C) -49
D) 38

Question 71

Find the indicated term.

-Which term of the arithmetic sequence $4,10,16, \ldots$ , is equal to 154 ?

A) Term 32
B) Term 27
C) Term 20
D) Term 26

Question 72

Find the indicated term.

-Which term of the arithmetic sequence $11,6,1, \ldots$ , is equal to - 489 ?

A) Term 96
B) Term 101
C) Term 91
D) Term 116

Question 73

Find the partial sum of the arithmetic sequence with the given first term and common difference

- $\mathrm{s}_{5}, \mathrm{a}_{1}=2, \mathrm{~d}=1$

A) 20
B) 7
C) 2
D) 10

Question 74

Find the partial sum of the arithmetic sequence with the given first term and common difference

- $\mathrm{s}_{6}, \mathrm{a} 1=-12, \mathrm{~d}=-2$

A) -12
B) -72
C) -24
D) -102

Question 75

Find the indicated partial sum for the given sequence

- $s_{6}, 5,7,9,11, \ldots$

A) 5
B) 30
C) 60
D) 17

Question 76

Find the indicated partial sum for the given sequence

- $\mathrm{s}_{4},-12,-10,-8,-6, \ldots$

A) -36
B) -4
C) -12
D) -48

Question 77

Find the indicated partial sum for the given sequence

-s10, $92,3,-86,-175, \ldots$

A) 3545
B) 4925
C) -3085
D) -3545

Question 78

Find the indicated partial sum for the given sequence

-s10, $1,-73,-147, \ldots$

A) 3325
B) -3325
C) -3320
D) 3340

Question 79

Find the indicated partial sum for the given sequence

- $s 4,81,-243,729,-2187, \ldots$

A) 4941
B) -2187
C) -1134
D) - 1620

Question 80

Find the indicated partial sum for the given sequence

-s8, $1,-5,9,-13, \ldots$

A) -15
B) 16
C) 5
D) -16