Deck 12: Further Topics in Algebra

Question 1

Write out the first five terms of the sequence.

- $a_{n}=n-5$

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

Answer 1

$-4,-3,-2,-1,0$

Question 2

Write out the first five terms of the sequence.

- $a_{n}=3 n-2$

A) $1,4,7,10,13$
B) $1,2,3,4,5$
C) $-1,-4,-7,-10,-13$
D) $5,8,11,14,17$

Answer 2

$1,4,7,10,13$

Question 3

Write out the first five terms of the sequence.

- $\mathrm{a}_{\mathrm{n}}=4^{\mathrm{n}}$

A) $1,16,81,256,625$
B) $1,4,16,64,256$
C) $16,64,256,1024,4096$
D) $4,16,64,256,1024$

Answer 3

$4,16,64,256,1024$

Question 4

Write out the first five terms of the sequence.

- $a_{n}=\frac{1}{n^{2}}$

A) $\frac{1}{4}, \frac{2}{9}, \frac{3}{16}, \frac{4}{25}, \frac{5}{36}$

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

C) $\frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}, \frac{1}{36}$

D) $1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}$

Question 5

Write out the first five terms of the sequence.

- $a_{n}=(-1)^{n-7}$

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 6

Write out the first five terms of the sequence.

- $a_{n}=\frac{n}{n+3}$

A) $\frac{1}{4}, \frac{2}{5}, \frac{1}{2}, \frac{4}{7}, \frac{5}{8}$

B) $\frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, 0$

C) $\frac{1}{4}, \frac{2}{5}, \frac{1}{2}, \frac{4}{7}, \frac{11}{7}$

D) $0, \frac{1}{4}, \frac{2}{5}, \frac{1}{2}, \frac{4}{7}$

Question 7

Write out the first five terms of the sequence.

- $a_{n}=\frac{(-1)^{n}}{n^{2}-2}$

A) $-\frac{1}{2}, 1, \frac{1}{2},-\frac{1}{7}, \frac{1}{14}$

B) $-1,-\frac{1}{2}, \frac{1}{7},-\frac{1}{14}, \frac{1}{23}$

C) $1, \frac{1}{2},-\frac{1}{7}, \frac{1}{14}, \frac{1}{23}$

D) $1, \frac{1}{2},-\frac{1}{7}, \frac{1}{14},-\frac{1}{23}$

Question 8

Write out the first five terms of the sequence.

- $a_{n}=n+\frac{1}{n}$

A) $2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \frac{6}{5}$

B) $1, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \frac{6}{5}$

C) $0,2, \frac{5}{2}, \frac{10}{3}, \frac{17}{4}$

D) $2, \frac{5}{2}, \frac{10}{3}, \frac{17}{4}, \frac{26}{5}$

Question 9

Write out the first five terms of the sequence.

- $a_{n}=n-\frac{4}{n}$

A) $-1, \frac{7}{2}, \frac{14}{3}, \frac{23}{4},-\frac{2}{5}$

B) $5,4, \frac{13}{3}, 5, \frac{29}{5}$

C) $-3,-1,-\frac{1}{3}, 0, \frac{1}{5}$

D) $-3,0, \frac{5}{3}, 3, \frac{21}{5}$

Question 10

Find the indicated term for the sequence.

- $a_{n}=4(4 n-1)$ ; $a_{8}$

A) 124
B) 128
C) 112
D) 136

Question 11

Find the indicated term for the sequence.

- $a_{n}=4 n-1 ; a_{8}$

A) 32
B) 33
C) 24
D) 31

Question 12

Find the indicated term for the sequence.

- $a_{n}=n^{2}-n$ ; ag

A) -72
B) 72
C) 90
D) 9

Question 13

Find the indicated term for the sequence.

- $\mathrm{a}_{\mathrm{n}}=4^{\mathrm{n}}$ ; $\mathrm{a}_{3}$

A) 64
B) 12
C) 16
D) 81

Question 14

Find the indicated term for the sequence.

- $a_{n}=(3 n-8)(2 n+8) ; a_{10}$

A) 616
B) 384
C) 494
D) 750

Question 15

Find the indicated term for the sequence.

- $\mathrm{a}_{\mathrm{n}}=\frac{2 \mathrm{n}-1}{3 \mathrm{n}+4} ; \mathrm{a}_{13}$

A) $\frac{25}{43}$

B) $\frac{23}{43}$

C) $\frac{5}{8}$

D) $\frac{23}{40}$

Question 16

Find a general term, $a_{n}$ , for the given terms of the sequence.

- $-4,-8,-12,-16,-20, \ldots$

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

Question 17

Find a general term, $a_{n}$ , for the given terms of the sequence.

- $6,14,22,30,38, \ldots$

A) $a_{n}=8 n-1$
B) $a_{n}=2 n-8$
C) $a_{n}=2(4 n-1)$
D) $a_{n}=6(8)^{n-1}$

Question 18

Find a general term, $a_{n}$ , for the given terms of the sequence.

- $1,4,7,10,13, \ldots$

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

Question 19

Find a general term, $a_{n}$ , for the given terms of the sequence.

- $0,2,6,12,20, \ldots$

A) $a_{n}=2^{n-1}-1$
B) $a_{n}=2 n-2$
C) $a_{n}=4 n-6$
D) $a_{n}=n^{2}-n$

Question 20

Find a general term, $a_{n}$ , for the given terms of the sequence.

- $1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}, \ldots$

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

Question 21

Find a general term, $a_{n}$ , for the given terms of the sequence.

- $\frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \frac{5}{16}, \frac{5}{32}, \ldots$

A) $a_{n}=\frac{5}{2 n}$

B) $a_{n}=\frac{2}{5^{n}}$

C) $a_{n}=\frac{5}{2^{n}}$

D) $a_{n}=\frac{2}{5 n}$

Question 22

Find a general term, $a_{n}$ , for the given terms of the sequence.

- $\frac{2}{7}, \frac{3}{8}, \frac{4}{9}, \frac{5}{10}, \ldots$

A) $a_{n}=\frac{n+1}{n+6}$

B) $a_{n}=\frac{n+1}{n^{6}}$

C) $a_{n}=\frac{n}{n-6}$

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

Question 23

A man earned $\$ 3500$ the first year he worked. If he received a raise of $\$ 500$ at the end of each year, what was his salary during the 15 th year?

A) $\$ 10,500$
B) $\$ 7000$
C) $\$ 11,000$
D) None of the above

Question 24

The population of a town was 27,000 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) 4500
B) 22,800
C) 22,200
D) 22,500

Question 25

An investment is worth $\$ 1000$ , and its value is increasing by $9 \%$ every year. What will its value be at the end of 9 years? Round your answer to the nearest dollar.

A) $\$ 2367$
B) $\$ 2172$
C) $\$ 1810$
D) \$1993

Question 26

An investment is worth $\$ 20,000$ , and its value is increasing by $10 \%$ every year. What will its value be at the end of 4 years? Round your answer to the nearest dollar.

A) $\$ 29,282$
B) $\$ 9282$
C) $\$ 28,000$
D) $\$ 26,620$

Question 27

Evaluate the series

- $\sum_{i=1}^{4}(2 \mathrm{i}-3)$

A) 8
B) 7
C) 9
D) 5

Question 28

Evaluate the series

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

A) 42
B) 17
C) 54
D) 38

Question 29

Evaluate the series

- $\sum_{\mathrm{i}=1}^{4} 3^{\mathrm{i}}$

A) 243
B) 12
C) 120
D) 81

Question 30

Evaluate the series

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

A) 39
B) 40
C) 78
D) 21

Question 31

Evaluate the series

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

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

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

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

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

Question 32

Evaluate the series

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

A) -22
B) -2
C) -18
D) 23

Question 33

Write out the terms of the series.

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

A) $(x+1)+(x+2)+(x+3)$
B) $(x+2)+(x+3)+(x+4)+(x+5)$
C) $x+(x+1)+(x+2)+(x+3)$
D) $(x+1)+(x+2)+(x+3)+(x+4)$

Question 34

Write out the terms of the series.

- $\sum_{\mathrm{i}=1}^{3}(\mathrm{x}+\mathrm{i})$

A) $(x+1)+(x+2)+(x+3)$
B) $(2 x+1)+(2 x+2)+(2 x+3)$
C) $(x+2)+(x+3)+(x+4)$
D) $(x+2)+(x+2)+(x+2)$

Question 35

Write out the terms of the series.

- $\sum_{i=1}^{5} i \cdot x^{i}$

A) $x+x^{2}+x^{3}+x^{4}+x^{5}$
B) $x+2 x^{2}+3 x^{3}+4 x^{4}+5 x^{5}+\ldots$
C) $1+2 x^{2}+3 x^{3}+4 x^{4}+5 x^{5}$
D) $x+2 x^{2}+3 x^{3}+4 x^{4}+5 x^{5}$

Question 36

Write out the terms of the series.

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

A) $-x-\frac{x^{2}}{4}-\frac{x^{3}}{9}-\frac{x^{4}}{16}$

B) $-x-\frac{x^{2}}{2}-\frac{x^{3}}{3}-\frac{x^{4}}{4}$

C) $-x-4 x^{2}-9 x^{3}-16 x^{4}$

D) $-x+\frac{x^{2}}{4}-\frac{x^{3}}{9}+\frac{x^{4}}{16}$

Question 37

Write the series with summation notation.

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

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

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

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

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

Question 38

Write the series with summation notation.

- $3+9+15+21+27$

A) $\sum_{i=1}^{5}(6 n-3)$

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

C) $\sum_{i=1}^{5}(6 i-3)$

D) $\sum_{\mathrm{i}=0}^{5}(6 \mathrm{i}-3)$

Question 39

Write the series with summation notation.

- $-2+3-4+5-6+7$

A) $\sum_{\mathrm{i}=1}^{6}(-1)^{\mathrm{i}} \cdot(\mathrm{i}+1)$

B) $\sum_{i=1}^{6}(-1)^{i+1} \cdot(i+1)$

C) $\sum_{i=1}^{6}(-1)^{\mathrm{i}+1} \cdot(\mathrm{i}+2)$

D) $\sum_{i=1}^{6}(-1)^{\mathrm{i}} \cdot(\mathrm{i}+2)$

Question 40

Find the arithmetic mean for the collection of numbers.
$7,13,3,15,12$

A) 51
B) 10
C) 9
D) 12.5

Question 41

Find the arithmetic mean for the collection of numbers. $9,6,14,20,11,5,19,4$

A) 9
B) 13
C) 10
D) 11

Question 42

Find the arithmetic mean for the collection of numbers.
$1,13,-3,4,15$

A) 13
B) 7.5
C) 4
D) 6

Question 43

Find the arithmetic mean for the collection of numbers.
$11,7,5,14,9,8,17$

A) $\frac{62}{7}$

B) $\frac{71}{7}$

C) $\frac{78}{7}$

D) 10

Question 44

The number of mutual funds available to investors in Dereguland for each year during the period $1996-2000$ is given in the following table.

To the nearest whole number, what was the average number of funds available during this period?

A) 5102
B) 5099
C) 12,393
D) 4957

Question 45

If the sequence is arithmetic, find the common difference. If the sequence is not arithmetic, say so.

- $7,9,11,13, \ldots$

A) 1.5
B) 2
C) 7
D) Not arithmetic

Question 46

If the sequence is arithmetic, find the common difference. If the sequence is not arithmetic, say so.

- $4,5,6,7, \ldots$

A) -1
B) 3
C) 1
D) Not arithmetic

Question 47

If the sequence is arithmetic, find the common difference. If the sequence is not arithmetic, say so.

- $-10,-12,-14,-16, \ldots$

A) -4
B) -6
C) -2
D) Not arithmetic

Question 48

If the sequence is arithmetic, find the common difference. If the sequence is not arithmetic, say so.

- $6.24,9.1,11.96,14.82, \ldots$

A) 2.86
B) 2.145
C) 8.58
D) Not arithmetic

Question 49

If the sequence is arithmetic, find the common difference. If the sequence is not arithmetic, say so.

- $4,8,16,32,64, \ldots$

A) $\frac{1}{2}$
B) 8
C) 2
D) Not arithmetic

Question 50

Write the first five terms of the arithmetic sequence.

- $a_{1}=6, d=4$

A) $0,6,10,14,18$
B) $6,9,12,15,18$
C) $6,10,14,18,22$
D) $10,14,18,22,27$

Question 51

Write the first five terms of the arithmetic sequence.

- $a_{1}=21, d=7$

A) $21,28,35,42,49$
B) $21,7,56,84,112$
C) $21,7,35,42,49$
D) $7,28,49,70,91$

Question 52

Write the first five terms of the arithmetic sequence.

- $\mathrm{a}_{1}=10, \mathrm{~d}=-2$

A) $-10,-8,-6,-4,-2$
B) $10,8,6,4,2$
C) $14,11,8,5$ , 2
D) $0,10,8,6,4$

Question 53

Write the first five terms of the arithmetic sequence.

- $a_{1}=7, d=-2$

A) $9,7,5,3,1$
B) $7,5,3,1,-1$
C) $7,5,2,1,-1$
D) $5,3,1,-1,-3$

Question 54

Write the first five terms of the arithmetic sequence.

- $\mathrm{a}_{1}=-24, \mathrm{~d}=7$

A) $-10,-17,-24,-31,-38$
B) $-24,-17,-10,-3,4$
C) $-10,-3,4,11,18$
D) $4,-3,-10,-17,-24$

Question 55

Write the first five terms of the arithmetic sequence.

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

A) $5,7,9,11,13$
B) $7,9,11,13,15$
C) $-7,-9,-11,-13,-15$
D) $-5,-7,-9,-11,-13$

Question 56

Use the formula for $a_{n}$ to find the general term for the arithmetic sequence.

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

A) $a_{n}=4 n+4$
B) $a_{n}=-4 n+8$
C) $a_{n}=4 n-4$
D) $a_{n}=-4 n-4$

Question 57

Use the formula for $a_{n}$ to find the general term for the arithmetic sequence.

- $2,-2,-6, \ldots$

A) $a_{n}=-4 n+6$
B) $a_{n}=2 n-4$
C) $\mathrm{a}_{\mathrm{n}}=\mathrm{n}-4$
D) $a_{n}=6 n-4$

Question 58

Use the formula for $a_{n}$ to find the general term for the arithmetic sequence.

- $a_{1}=-4, d=\frac{5}{3}$

A) $a_{n}=\frac{5}{3}+\frac{7}{3} n$

B) $a_{n}=\frac{5}{3} n-\frac{17}{3}$

C) $a_{n}=\frac{5}{3} n-\frac{7}{3}$

D) $a_{n}=4 n-\frac{7}{3}$

Question 59

Use the formula for $a_{n}$ to find the general term for the arithmetic sequence.

- $-14,-10,-6,-2,2$

A) $a_{n}=5 n-19$
B) $\mathrm{a}_{\mathrm{n}}=4 \mathrm{n}-18$
C) $a_{n}=5 n+18$
D) $a_{n}=18 n-4$

Question 60

Find the indicated term for the arithmetic sequence.

- $a_{1}=-2, d=3$ ; $a_{8}$

A) -23
B) 22
C) -26
D) 19

Question 61

Find the indicated term for the arithmetic sequence.

- $a_{1}=5, d=-2$ ; $a_{39}$

A) 81
B) 83
C) -73
D) -71

Question 62

Find the indicated term for the arithmetic sequence.

- $\mathrm{a}_{1}=-7, \mathrm{~d}=\frac{7}{8} ; \mathrm{a}_{49}$

A) $\frac{287}{8}$

B) -49

C) $-\frac{399}{8}$

D) 35

Question 63

Find the indicated term for the arithmetic sequence.

- $6,4,2, \ldots$ ; a22

A) 48
B) 50
C) -36
D) -38

Question 64

Find the indicated term for the arithmetic sequence.

- $\mathrm{a}_{11}=24, \mathrm{a}_{16}=34 ; \mathrm{a}_{1}$

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

Question 65

Find the indicated term for the arithmetic sequence.

- $\mathrm{a}_{11}=-41, \mathrm{a}_{19}=-81 ; \mathrm{a}_{7}$

A) -26
B) -5
C) -21
D) 9

Question 66

Find the indicated term for the arithmetic sequence.

- $\mathrm{a}_{40}=\frac{204}{5}, \mathrm{a}_{50}=\frac{254}{5} ; \mathrm{a}_{5}$

A) 9

B) 1

C) $\frac{29}{5}$

D) $\frac{34}{5}$

Question 67

Find the number of terms in the arithmetic sequence.

- $9,17,25, \ldots, 105$

A) 13
B) 14
C) 11
D) 12

Question 68

Find the number of terms in the arithmetic sequence.

- $-7,1,9, \ldots, 177$

A) 24
B) 25
C) 23
D) 22

Question 69

Find the number of terms in the arithmetic sequence.

- $8,-1,-10, \ldots,-109$

A) 12
B) 15
C) 14
D) 13

Question 70

Find the number of terms in the arithmetic sequence.

- $3, \frac{5}{2}, 2, \ldots,-4$

A) 13
B) 16
C) 15
D) 14

Question 71

Find the requested sum for the arithmetic sequence.

- $\mathrm{a}_{1}=1, \mathrm{~d}=-5 ; \mathrm{S}_{6}$

A) 6
B) -29
C) -24
D) -69

Question 72

Find the requested sum for the arithmetic sequence.

- $\mathrm{a}_{1}=-15, \mathrm{~d}=-1 ; \mathrm{S}_{4}$

A) -66
B) -18
C) -19
D) -60

Question 73

Find the requested sum for the arithmetic sequence.

- $a_{n}=-5 n+8 ; S_{5}$

A) -17
B) -35
C) 15
D) -23

Question 74

Find the requested sum for the arithmetic sequence.

- $a_{n}=-3 n-9 ; S_{5}$

A) -60
B) 0
C) -90
D) -24

Question 75

Use a formula for $S_{\mathbf{n}}$ to evaluate the series.

- $\sum_{i=1}^{4}(5 i-4)$

A) 16
B) 32
C) 34
D) 4

Question 76

Use a formula for $S_{\mathbf{n}}$ to evaluate the series.

- $\sum_{\mathrm{i}=1}^{6}\left(-\frac{5}{2} \mathrm{i}+3\right)$

A) $-\frac{69}{2}$

B) -12

C) -36

D) $-\frac{23}{2}$

Question 77

Use a formula for $S_{\mathbf{n}}$ to evaluate the series.

- $\sum_{i=1}^{7532} i$

A) 7533
B) $28,361,746$
C) $28,369,278$
D) 7531

Question 78

An auditorium has 25 rows, with 10 seats in the first row, 12 in the second row, 14 in the third row, and so forth. How many seats are in the auditorium?

A) 850
B) 900
C) 550
D) 500

Question 79

If a person puts one penny in a piggy bank on the first day, two pennies in on the second day, three pennies in on the third day, and so forth, how much money will be in the bank after 30 days?

A) $\$ 9.30$
B) $\$ 0.30$
C) $\$ 4.65$
D) $\$ 2.33$

Question 80

A collection of dimes is arranged in a triangular array, with 14 coins in the base row, 13 in the next, 12 in the next, and so forth. Find the value of the collection.

A) $\$ 10.50$
B) $\$ 1.05$
C) $\$ 5.25$
D) $\$ 21.00$