Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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.

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

Answers may vary. One possibility: \[\be

Question 11

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

Accepted Answer

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)  $2 , - \frac { 1 } { 2 } , 2 , - \frac { 1 } { 2 }$ 
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 12

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

Accepted Answer

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

Question 13

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.

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

Find the indicated term of the sequence.
-$$a _ { n } = 2 n - 2 ; a _ { 10 }$$

Accepted Answer

A)  20 
B)  0 
C)  18 
D)  22 
A)  20 
B)  0 
C)  18 
D)  22

Question 18

Solve.
-Suppose 6 cards are drawn from a deck of 52 cards. What is the probability of drawing 3 spades and 3 hearts?

Accepted Answer

A)  $\frac { 1296 } { 72,709 }$ 
B)  $\frac { 1573 } { 391,510 }$ 
C)  $\frac { 3 } { 5,089,630 }$ 
D)  $\frac { 1573 } { 49,980 }$ 
A)  $\frac { 1296 } { 72,709 }$ 
B)  $\frac { 1573 } { 391,510 }$ 
C)  $\frac { 3 } { 5,089,630 }$ 
D)  $\frac { 1573 } { 49,980 }$

Question 19

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

Accepted Answer

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

Question 20

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

What are the values of c and f?

Accepted Answer

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

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

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

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

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

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

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

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

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.

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

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

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

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

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.

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

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

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

Find the indicated term of the sequence. - $a _ { n } = 2 n - 2 ; a _ { 10 }$

Solve. -Suppose 6 cards are drawn from a deck of 52 cards. What is the probability of drawing 3 spades and 3 hearts?

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

Solve the problem. -An experiment was conducted. The results are listed in the following table. What are the values of c and f?

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Exam 8: Sequences, Series, and Combinatorics

Find fraction notation. - 0.858585‾0.8585 \overline { 85 }0.858585

Solve the problem. - 46P1046 ^ { \mathrm { P } } 1046P10

Find the sum, if it exists. - ∑i=1∞10(−52)i−1\sum _ { i = 1 } ^ { \infty } 10 \left( - \frac { 5 } { 2 } \right) ^ { i - 1 }∑i=1∞​10(−25​)i−1

Evaluate. - (125)\left( \begin{array} { c } 12 \\ 5 \end{array} \right)(125​)

Find the sum. - ∑i=16(5i−4)\sum _ { i = 1 } ^ { 6 } ( 5 i - 4 )∑i=16​(5i−4)

Solve the problem. - 20C420 C _ { 4 }20C4​

Evaluate. - 5!3!\frac { 5 ! } { 3 ! }3!5!​

Solve. -A pendulum bob swings 5.0 cm5.0 \mathrm {~cm}5.0 cm on its first oscillation. On each subsequent oscillation the bob travels 25\frac { 2 } { 5 }52​ of the previous distance. Find the total distance the bob travels before coming to rest.

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

Use mathematical induction to prove the following. - 1⋅2+2⋅3+3⋅4+…+n(n+1)=n(n+1)(n+2)31 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }1⋅2+2⋅3+3⋅4+…+n(n+1)=3n(n+1)(n+2)​

Find the first 4 terms of the recursively defined sequence. - a1=2,an+1=(−1)nana _ { 1 } = 2 , a _ { n } + 1 = \frac { ( - 1 ) ^ { n } } { a _ { n } }a1​=2,an​+1=an​(−1)n​

Evaluate the sum. - ∑i=053i\sum _ { i = 0 } ^ { 5 } 3 ^ { i }∑i=05​3i

Determine how many of the first 5 statements in the sequence obtainable from the given statement are true. - n2+2n+1n ^ { 2 } + 2 n + 1n2+2n+1 is a perfect square.

Find the indicated quantity. - a1a _ { 1 }a1​ , when d=−3d = - 3d=−3 and a36=−93a _ { 36 } = - 93a36​=−93

Evaluate the sum. - ∑k=010(−1)k\sum _ { \mathrm { k } = 0 } ^ { 10 } ( - 1 ) ^ { \mathrm { k } }∑k=010​(−1)k

Evaluate the sum. - ∑k=14(−1)k(k+3)\sum _ { \mathrm { k } = 1 } ^ { 4 } ( - 1 ) ^ { \mathrm { k } } ( \mathrm { k } + 3 )∑k=14​(−1)k(k+3)

Find the indicated term of the sequence. - an=2n−2;a10a _ { n } = 2 n - 2 ; a _ { 10 }an​=2n−2;a10​