Question 1

Write the sum using summation notation, assuming the suggested pattern continues.
-$$36 + 49 + 64 + 81 + \ldots + n ^ { 2 } + \ldots$$

Accepted Answer

A)  $\sum _ { n = 7 } ^ { \infty } n ^ { 2 }$ 
B)  $\sum _ { n = 6 } ^ { \infty } n - 1 ^ { 2 }$ 
C)  $\sum _ { n = 6 } ^ { \infty } n ^ { 2 }$ 
D)  $\sum _ { n = 6 } ^ { \infty } n + 1 ^ { 2 }$ 
A)  $\sum _ { n = 7 } ^ { \infty } n ^ { 2 }$ 
B)  $\sum _ { n = 6 } ^ { \infty } n - 1 ^ { 2 }$ 
C)  $\sum _ { n = 6 } ^ { \infty } n ^ { 2 }$ 
D)  $\sum _ { n = 6 } ^ { \infty } n + 1 ^ { 2 }$

Question 2

Find the sum.
-$$1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 4 ^ { 2 } + \ldots + 87 ^ { 2 }$$

Accepted Answer

A)  $220,762.5$ 
B)  112,288 
C)  223,300 
D)  $1,339,800$ 
A)  $220,762.5$ 
B)  112,288 
C)  223,300 
D)  $1,339,800$

Question 3

For the given statement $ P_{n} $, write the statements $ P_{1}, P_{k} $, and $ P_{k+1} $
-$$2 + 4 + 8 + \ldots + 2 n = 2^{ n + 1} - 2$$

Accepted Answer

The answer of For the given statement $ P_{n} $,...

Question 4

State an explicit rule for the nth term of the recursively-defined sequence.
-$$a _ { n } = a _ { n - 1 } - 10 ; a _ { 1 } = 1$$

Accepted Answer

A)  $a _ { n } = 10 - 1 ( n )$ 
B)  $a _ { n } = 1 - 10 ( n - 1 )$ 
C)  $a _ { n } = 1 - 10 ( n )$ 
D)  $a _ { n } = 10 - 1 ( n - 1 )$ 
A)  $a _ { n } = 10 - 1 ( n )$ 
B)  $a _ { n } = 1 - 10 ( n - 1 )$ 
C)  $a _ { n } = 1 - 10 ( n )$ 
D)  $a _ { n } = 10 - 1 ( n - 1 )$

Question 5

Tell whether permutations or combinations are being described.
-4 musicians are selected to form a band.

Accepted Answer

A)  Permutations 
B)  Combinations 
A)  Permutations 
B)  Combinations

Question 6

Use mathematical induction to prove the statement is true for all positive integers n.
-$$\left( 1 - \frac { 1 } { 2 } \right) \left( 1 - \frac { 1 } { 3 } \right) \ldots \left( 1 - \frac { 1 } { n + 1 } \right) = \frac { 1 } { n + 1 }$$

Accepted Answer

To prove the given statement using mathe

Question 7

Expand the binomial.
-$$( 2 x + 4 ) ^ { 3 }$$

Accepted Answer

A)  $8 x ^ { 3 } + 48 x ^ { 2 } + 48 x + 64$ 
B)  $4 x ^ { 6 } + 8 x ^ { 3 } + 4096$ 
C)  $4 x ^ { 2 } + 16 x + 16$ 
D)  $8 x ^ { 3 } + 48 x ^ { 2 } + 96 x + 64$ 
A)  $8 x ^ { 3 } + 48 x ^ { 2 } + 48 x + 64$ 
B)  $4 x ^ { 6 } + 8 x ^ { 3 } + 4096$ 
C)  $4 x ^ { 2 } + 16 x + 16$ 
D)  $8 x ^ { 3 } + 48 x ^ { 2 } + 96 x + 64$

Question 8

Find a recursive rule for the nth term of the sequence.
-$$- 10,2,14,26 , \ldots$$

Accepted Answer

A)  $a _ { n } = a _ { n - 1 } + 13$ 
B)  $a _ { n } = a _ { n - 1 } - 12$ 
C)  $a _ { n } = a _ { n - 1 } - 13$ 
D)  $a _ { n } = a _ { n - 1 } + 12$ 
A)  $a _ { n } = a _ { n - 1 } + 13$ 
B)  $a _ { n } = a _ { n - 1 } - 12$ 
C)  $a _ { n } = a _ { n - 1 } - 13$ 
D)  $a _ { n } = a _ { n - 1 } + 12$

Question 9

Find an explicit rule for the nth term of the sequence.
-The second and fifth terms of a geometric sequence are 9 and 243 , respectively.

Accepted Answer

A)  $a _ { n } = 3 \cdot 3 ^ { n + 3 }$ 
B)  $a _ { n } = 3 \cdot 3 ^ { n - 2 }$ 
C)  $a _ { n } = 3 \cdot 3 ^ { n - 1 }$ 
D)  $a _ { n } = 3 \cdot 3 ^ { n + 1 }$ 
A)  $a _ { n } = 3 \cdot 3 ^ { n + 3 }$ 
B)  $a _ { n } = 3 \cdot 3 ^ { n - 2 }$ 
C)  $a _ { n } = 3 \cdot 3 ^ { n - 1 }$ 
D)  $a _ { n } = 3 \cdot 3 ^ { n + 1 }$

Question 10

Solve.
-A collection of dimes is arranged in a triangular array with 17 coins in the base row, 16 in the next, 15 in the next, and so forth with 1 dime in the last row. Find the value of the collection.

Accepted Answer

A)  $\$ 7.65$ 
B)  $\$ 1.53$ 
C)  $\$ 30.60$ 
D)  $\$ 15.30$ 
A)  $\$ 7.65$ 
B)  $\$ 1.53$ 
C)  $\$ 30.60$ 
D)  $\$ 15.30$

Question 11

Evaluate.
-${ } _ { 5 } \mathrm { P } _ { 1 }$

Accepted Answer

A)  48 
B)  5 
C)  120 
D)  $2.5$ 
A)  48 
B)  5 
C)  120 
D)  $2.5$

Question 12

Find a recursive rule for the nth term of the sequence.
-$$- 7 , - 42 , - 252 , - 1512 , \ldots$$

Accepted Answer

A)  $a _ { n } = a _ { n - 1 } \cdot 12$ 
B)  $a _ { n } = a _ { n - 1 } + 6$ 
C)  $a _ { n } = a _ { n - 1 } + 12$ 
D)  $a _ { n } = a _ { n - 1 } \cdot 6$ 
A)  $a _ { n } = a _ { n - 1 } \cdot 12$ 
B)  $a _ { n } = a _ { n - 1 } + 6$ 
C)  $a _ { n } = a _ { n - 1 } + 12$ 
D)  $a _ { n } = a _ { n - 1 } \cdot 6$

Question 13

Find the sum of the arithmetic sequence.
-2, -4, 8, -16, 32

Accepted Answer

A)  62 
B)  22 
C)  -62 
D)  -22 
A)  62 
B)  22 
C)  -62 
D)  -22

Question 14

Determine whether the sequence converges or diverges. If it converges, give the limit.
-$50 , \frac { 25 } { 2 } , \frac { 25 } { 8 } , \frac { 25 } { 32 } , \ldots$

Accepted Answer

A)  Converges; $\frac { 200 } { 3 }$ 
B)  Converges; $- 4250$ 
C)  Diverges 
D)  Converges; 0 
A)  Converges; $\frac { 200 } { 3 }$ 
B)  Converges; $- 4250$ 
C)  Diverges 
D)  Converges; 0

Question 15

Find the sum.
-$14 + 16 + 18 + 20 + \ldots + 314$

Accepted Answer

A)  24,764 
B)  25,080 
C)  24,450 
D)  47,414 
A)  24,764 
B)  25,080 
C)  24,450 
D)  47,414

Question 16

Evaluate.
-${ } _ { 11 } \mathrm { C } _ { 7 }$

Accepted Answer

A)  48 
B)  330 
C)  831,600 
D)  7920 
A)  48 
B)  330 
C)  831,600 
D)  7920

Question 17

Use mathematical induction to prove the statement is true for all positive integers n.
-$$3 + 2 \cdot 3 + 3 \cdot 3 + \ldots + 3 n = \frac { 3 n ( n + 1 ) } { 2 }$$

Accepted Answer

To prove the given statement using mathe

Question 18

Construct a graph for the first ten terms of the sequence.
-$$a _ { n } = \sqrt { n } - \sqrt { n - 1 }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 19

Solve.
-A musician plans to perform 5 selections for a concert. If he can choose from 7 different selections, how many ways can he arrange his program?

Accepted Answer

A)  21 
B)  35 
C)  2520 
D)  16,807 
A)  21 
B)  35 
C)  2520 
D)  16,807

Question 20

Use mathematical induction to prove the statement is true for all positive integers n.
-$$5 + 5 \cdot \frac { 1 } { 7 } + 5 \cdot \left( \frac { 1 } { 7 } \right) ^ { 2 } + \ldots + 5 \cdot \left( \frac { 1 } { 7 } \right) ^ { n - 1 } = \frac { 5 \left( 1 - \left( \frac { 1 } { 7 } \right) ^ { n } \right) } { 1 - \frac { 1 } { 7 } }$$

Accepted Answer

To prove the given statement using mathe

Write the sum using summation notation, assuming the suggested pattern continues. - $36 + 49 + 64 + 81 + \ldots + n ^ { 2 } + \ldots$

Find the sum. - $1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 4 ^ { 2 } + \ldots + 87 ^ { 2 }$

For the given statement $P_{n}$ , write the statements $P_{1}, P_{k}$ , and $P_{k+1}$ - $2 + 4 + 8 + \ldots + 2 n = 2^{ n + 1} - 2$

State an explicit rule for the nth term of the recursively-defined sequence. - $a _ { n } = a _ { n - 1 } - 10 ; a _ { 1 } = 1$

Tell whether permutations or combinations are being described. -4 musicians are selected to form a band.

Use mathematical induction to prove the statement is true for all positive integers n. - $\left( 1 - \frac { 1 } { 2 } \right) \left( 1 - \frac { 1 } { 3 } \right) \ldots \left( 1 - \frac { 1 } { n + 1 } \right) = \frac { 1 } { n + 1 }$

Expand the binomial. - $( 2 x + 4 ) ^ { 3 }$

Find a recursive rule for the nth term of the sequence. - $- 10,2,14,26 , \ldots$

Find an explicit rule for the nth term of the sequence. -The second and fifth terms of a geometric sequence are 9 and 243 , respectively.

Solve. -A collection of dimes is arranged in a triangular array with 17 coins in the base row, 16 in the next, 15 in the next, and so forth with 1 dime in the last row. Find the value of the collection.

Evaluate. - ${ } _ { 5 } \mathrm { P } _ { 1 }$

Find a recursive rule for the nth term of the sequence. - $- 7 , - 42 , - 252 , - 1512 , \ldots$

Find the sum of the arithmetic sequence. -2, -4, 8, -16, 32

Determine whether the sequence converges or diverges. If it converges, give the limit. - $50 , \frac { 25 } { 2 } , \frac { 25 } { 8 } , \frac { 25 } { 32 } , \ldots$

Find the sum. - $14 + 16 + 18 + 20 + \ldots + 314$

Evaluate. - ${ } _ { 11 } \mathrm { C } _ { 7 }$

Use mathematical induction to prove the statement is true for all positive integers n. - $3 + 2 \cdot 3 + 3 \cdot 3 + \ldots + 3 n = \frac { 3 n ( n + 1 ) } { 2 }$

Construct a graph for the first ten terms of the sequence. - $a _ { n } = \sqrt { n } - \sqrt { n - 1 }$ $Construct a graph for the first ten terms of the sequence. - a _ { n } = \sqrt { n } - \sqrt { n - 1 }$

Solve. -A musician plans to perform 5 selections for a concert. If he can choose from 7 different selections, how many ways can he arrange his program?

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Exam 9: Discrete Mathematics

Write the sum using summation notation, assuming the suggested pattern continues. - 36+49+64+81+…+n2+…36 + 49 + 64 + 81 + \ldots + n ^ { 2 } + \ldots36+49+64+81+…+n2+…

Find the sum. - 12+22+32+42+…+8721 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 4 ^ { 2 } + \ldots + 87 ^ { 2 }12+22+32+42+…+872

For the given statement Pn P_{n} Pn​ , write the statements P1,Pk P_{1}, P_{k} P1​,Pk​ , and Pk+1 P_{k+1} Pk+1​ - 2+4+8+…+2n=2n+1−22 + 4 + 8 + \ldots + 2 n = 2^{ n + 1} - 22+4+8+…+2n=2n+1−2

State an explicit rule for the nth term of the recursively-defined sequence. - an=an−1−10;a1=1a _ { n } = a _ { n - 1 } - 10 ; a _ { 1 } = 1an​=an−1​−10;a1​=1

Tell whether permutations or combinations are being described. -4 musicians are selected to form a band.

Expand the binomial. - (2x+4)3( 2 x + 4 ) ^ { 3 }(2x+4)3

Find a recursive rule for the nth term of the sequence. - −10,2,14,26,…- 10,2,14,26 , \ldots−10,2,14,26,…

Find an explicit rule for the nth term of the sequence. -The second and fifth terms of a geometric sequence are 9 and 243 , respectively.

Solve. -A collection of dimes is arranged in a triangular array with 17 coins in the base row, 16 in the next, 15 in the next, and so forth with 1 dime in the last row. Find the value of the collection.

Evaluate. - 5P1{ } _ { 5 } \mathrm { P } _ { 1 }5​P1​

Find a recursive rule for the nth term of the sequence. - −7,−42,−252,−1512,…- 7 , - 42 , - 252 , - 1512 , \ldots−7,−42,−252,−1512,…

Find the sum of the arithmetic sequence. -2, -4, 8, -16, 32

Determine whether the sequence converges or diverges. If it converges, give the limit. - 50,252,258,2532,…50 , \frac { 25 } { 2 } , \frac { 25 } { 8 } , \frac { 25 } { 32 } , \ldots50,225​,825​,3225​,…

Find the sum. - 14+16+18+20+…+31414 + 16 + 18 + 20 + \ldots + 31414+16+18+20+…+314

Evaluate. - 11C7{ } _ { 11 } \mathrm { C } _ { 7 }11​C7​

Use mathematical induction to prove the statement is true for all positive integers n. - 3+2⋅3+3⋅3+…+3n=3n(n+1)23 + 2 \cdot 3 + 3 \cdot 3 + \ldots + 3 n = \frac { 3 n ( n + 1 ) } { 2 }3+2⋅3+3⋅3+…+3n=23n(n+1)​

Construct a graph for the first ten terms of the sequence. - an=n−n−1a _ { n } = \sqrt { n } - \sqrt { n - 1 }an​=n​−n−1​

Solve. -A musician plans to perform 5 selections for a concert. If he can choose from 7 different selections, how many ways can he arrange his program?

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Write the sum using summation notation, assuming the suggested pattern continues. - $36 + 49 + 64 + 81 + \ldots + n ^ { 2 } + \ldots$

Find the sum. - $1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 4 ^ { 2 } + \ldots + 87 ^ { 2 }$

For the given statement $P_{n}$ , write the statements $P_{1}, P_{k}$ , and $P_{k+1}$ - $2 + 4 + 8 + \ldots + 2 n = 2^{ n + 1} - 2$

State an explicit rule for the nth term of the recursively-defined sequence. - $a _ { n } = a _ { n - 1 } - 10 ; a _ { 1 } = 1$

Expand the binomial. - $( 2 x + 4 ) ^ { 3 }$

Find a recursive rule for the nth term of the sequence. - $- 10,2,14,26 , \ldots$

Evaluate. - ${ } _ { 5 } \mathrm { P } _ { 1 }$

Find a recursive rule for the nth term of the sequence. - $- 7 , - 42 , - 252 , - 1512 , \ldots$

Determine whether the sequence converges or diverges. If it converges, give the limit. - $50 , \frac { 25 } { 2 } , \frac { 25 } { 8 } , \frac { 25 } { 32 } , \ldots$

Find the sum. - $14 + 16 + 18 + 20 + \ldots + 314$

Evaluate. - ${ } _ { 11 } \mathrm { C } _ { 7 }$

Use mathematical induction to prove the statement is true for all positive integers n. - $3 + 2 \cdot 3 + 3 \cdot 3 + \ldots + 3 n = \frac { 3 n ( n + 1 ) } { 2 }$

Construct a graph for the first ten terms of the sequence. - $a _ { n } = \sqrt { n } - \sqrt { n - 1 }$ $Construct a graph for the first ten terms of the sequence. - a _ { n } = \sqrt { n } - \sqrt { n - 1 }$