Question 1

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

Accepted Answer

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)  $\left( \frac { 1 } { 7 } \right) ^ { n - 1 } + \frac { 8 } { 7 }$ 
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 2

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

Accepted Answer

Answers may vary. One possibility: @#LAT-DLM& : If

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

A) 8 
B) 2 
C) -2 $$\text { 
D)  } \frac { 1 } { 2 }$$ 
A) 8 
B) 2 
C) -2 $$\text { 
D)  } \frac { 1 } { 2 }$$

Question 6

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

Accepted Answer

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

Question 7

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?

Accepted Answer

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

Question 8

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

Accepted Answer

Answers may vary. One possibility: \[\be

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

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?

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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 $180 - \frac { 360 } { n }$. $( n \geq 3 )$

Accepted Answer

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

Question 15

Solve.
-How many 7-card hands are possible with a 52-card deck?

Accepted Answer

A)  $1.338 \times 10 ^ { 8 }$ 
B)  $9.365 \times 10 ^ { 8 }$ 
C)  $2,908,360$ 
D)  $18,009,460$ 
A)  $1.338 \times 10 ^ { 8 }$ 
B)  $9.365 \times 10 ^ { 8 }$ 
C)  $2,908,360$ 
D)  $18,009,460$

Question 16

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

Accepted Answer

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

Question 17

Find the indicated quantity.
-$\mathrm { n }$, when $\mathrm { a } _ { 1 } = 309 , \mathrm { a } _ { \mathrm { n } } = 49$, and $\mathrm { d } = - 20$;

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

Answers may vary. One possibility: \[\be

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

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

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

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

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

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

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?

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

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

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

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

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?

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

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 $180 - \frac { 360 } { n }$ . $( n \geq 3 )$

Solve. -How many 7-card hands are possible with a 52-card deck?

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

Find the indicated quantity. - $\mathrm { n }$ , when $\mathrm { a } _ { 1 } = 309 , \mathrm { a } _ { \mathrm { n } } = 49$ , and $\mathrm { d } = - 20$ ;

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

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

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

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Exam 8: Sequences, Series, and Combinatorics

Find the nth, or general, term. - 17,97,817,…\frac { 1 } { 7 } , \frac { 9 } { 7 } , \frac { 81 } { 7 } , \ldots71​,79​,781​,…

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

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

Evaluate. - 7P0{ } _ { 7 } \mathrm { P } _ { 0 }7​P0​

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

Evaluate. - 4!2!2!\frac { 4 ! } { 2 ! 2 ! }2!2!4!​