Question 1

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { n = 1 } ^ { 16 } n$$ ​

Accepted Answer

A) 18,496 
B) 8,976 
C) 1,496 
D) 272 
E) 136 
A) 18,496 
B) 8,976 
C) 1,496 
D) 272 
E) 136

Question 2

Determine whether the statement is true or false. ​A sequence with terms has n-1 second differences.
​

Accepted Answer

A) False 
B) True 
A) False 
B) True

Question 3

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { n = 1 } ^ { 2 } n ^ { 4 }$$ ​

Accepted Answer

A) 33 
B) 17 
C) 30 
D) 5 
E) 9 
A) 33 
B) 17 
C) 30 
D) 5 
E) 9

Question 4

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​
A0 = 1
An = an - 1 + n
​

Accepted Answer

A) 0,1,2,-4,7,-11 First differences: 1,1,2,3,4
Second differences: -1,1,-1,1
Neither 
B) 1,-2,4,-7,11,-16 First differences: -1,1,-1,1
Second differences: 1,2,3,4,5
Neither 
C) 0,1,2,4,7,11 First differences: 1,1,2,3,4
Second differences: -1,1,-1,1
Linear 
D) 0,1,2,4,7,11 First differences: 1,1,2,3,4
Second differences: -1,1,-1,1
Quadratic 
E) 1,2,4,7,11,16 First differences: 1,2,3,4,5
Second differences: 1,1,1,1
Quadratic 
A) 0,1,2,-4,7,-11 First differences: 1,1,2,3,4
Second differences: -1,1,-1,1
Neither 
B) 1,-2,4,-7,11,-16 First differences: -1,1,-1,1
Second differences: 1,2,3,4,5
Neither 
C) 0,1,2,4,7,11 First differences: 1,1,2,3,4
Second differences: -1,1,-1,1
Linear 
D) 0,1,2,4,7,11 First differences: 1,1,2,3,4
Second differences: -1,1,-1,1
Quadratic 
E) 1,2,4,7,11,16 First differences: 1,2,3,4,5
Second differences: 1,1,1,1
Quadratic

Question 5

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 12 } 9 n - 3 n ^ { 2 }$

Accepted Answer

A) -924 
B) -1248 
C) 468 
D) -324 
E) -3744 
A) -924 
B) -1248 
C) 468 
D) -324 
E) -3744

Question 6

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { n = 1 } ^ { 6 } n ^ { 2 }$$ ​

Accepted Answer

A) 21 
B) 42 
C) 546 
D) 441 
E) 91 
A) 21 
B) 42 
C) 546 
D) 441 
E) 91

Question 7

Use mathematical induction to solve for all positive integers n.​ $3 + 8 + 13 + 18 + \ldots + ( 5 n - 2 ) = ?$ ​

Accepted Answer

A)  $\frac { n } { 6 } ( 5 n + 1 )$ 
B)  $n ( 5 n + 1 )$ 
C)  $\frac { n } { 2 } ( 5 n + 1 )$ 
D)  $\frac { n } { 4 } ( 5 n + 1 )$ 
E)  $5 n + 1$ 
A)  $\frac { n } { 6 } ( 5 n + 1 )$ 
B)  $n ( 5 n + 1 )$ 
C)  $\frac { n } { 2 } ( 5 n + 1 )$ 
D)  $\frac { n } { 4 } ( 5 n + 1 )$ 
E)  $5 n + 1$

Question 8

The table shows the numbers $a _ { n }$ (in thousands)of residents from 2002 through 2007. $$\begin{array} { | c | l | } 
\hline  & \text { Number of residents, } \
&.\
{ \text { Year } }& a _ { n } \
&.\
\hline 2002 & 646 \
\hline 2003 & 657 \
\hline 2004 & 668 \
\hline 2005 & 679 \
\hline
\end{array}$$ $$\begin{array} { | l | l | } 
\hline 2006 & 690 \
\hline 2007 & 701 \
\hline
\end{array}$$
Determine whether a linear model can be used to approximate the data.
If so,find a model algebraically.Let n represent the year,with $n = 2$ corresponding to 2002.

Accepted Answer

A) A linear model can be used. $a _ { n } = 11 n + 690$ 
B) A linear model can be used. $a _ { n } = 11 n + 657$ 
C) A linear model can be used. $a _ { n } = 11 n + 701$ 
D) A linear model can be used. $a _ { n } = 11 n + 624$ 
E) A linear model can be used. $a _ { n } = 11 n + 646$ 
A) A linear model can be used. $a _ { n } = 11 n + 690$ 
B) A linear model can be used. $a _ { n } = 11 n + 657$ 
C) A linear model can be used. $a _ { n } = 11 n + 701$ 
D) A linear model can be used. $a _ { n } = 11 n + 624$ 
E) A linear model can be used. $a _ { n } = 11 n + 646$

Question 9

Find pS1U1B1k+1S1U1B0 for the given pk.? $$p _ { k } = \frac { k ^ { 2 } } { 7 ( k + 2 ) ^ { 2 } }$$ ?

Accepted Answer

A)  $$p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 3 ( k + 8 ) ^ { 2 } }$$ 
B)  $p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 8 ( k + 8 ) ^ { 2 } }$ 
C)  $$p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 8 ( k + 3 ) ^ { 2 } }$$ 
D)  $p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 7 ( k + 3 ) ^ { 2 } }$ 
E)  $p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 7 ( k + 8 ) ^ { 2 } }$ 
A)  $$p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 3 ( k + 8 ) ^ { 2 } }$$ 
B)  $p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 8 ( k + 8 ) ^ { 2 } }$ 
C)  $$p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 8 ( k + 3 ) ^ { 2 } }$$ 
D)  $p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 7 ( k + 3 ) ^ { 2 } }$ 
E)  $p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } } { 7 ( k + 8 ) ^ { 2 } }$

Question 10

Find a formula for the sum of the n terms of the sequence. $$\frac { 1 } { 2 } , \frac { 5 } { 4 } , \frac { 25 } { 8 } , \frac { 125 } { 16 } , \ldots$$

Accepted Answer

A)  $\frac { 5 \left( 5 ^ { n } - 2 ^ { n } \right) } { 3 \left( 2 ^ { n } \right) }$ 
B) ​ $\frac { 5 ^ { n - 1 } } { 2 }$ 
C)  $\frac { 5 ^ { n } + 2 ^ { n } } { 7 \left( 2 ^ { n } \right) }$ 
D)  $\frac { 1 } { 2 ^ { n } }$ 
E)  $\frac { 5 ^ { n } - 2 ^ { n } } { 3 \left( 2 ^ { n } \right) }$ 
A)  $\frac { 5 \left( 5 ^ { n } - 2 ^ { n } \right) } { 3 \left( 2 ^ { n } \right) }$ 
B) ​ $\frac { 5 ^ { n - 1 } } { 2 }$ 
C)  $\frac { 5 ^ { n } + 2 ^ { n } } { 7 \left( 2 ^ { n } \right) }$ 
D)  $\frac { 1 } { 2 ^ { n } }$ 
E)  $\frac { 5 ^ { n } - 2 ^ { n } } { 3 \left( 2 ^ { n } \right) }$

Question 11

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither.​ $\begin{array} { l } 
a _ { 0 } = 4 \
a _ { n } = \left( a _ { n - 1 } \right) ^ { 2 }
\end{array}$ ​

Accepted Answer

A) 0,4,-16,256,-65,536,4,294,967,296 First differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Second differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Linear 
B) 0,4,16,256,65,536,4,294,967,296 First differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Second differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Quadratic 
C) 4,-16,256,-65,536,4,294,967,296,-18,446,744,073,709,600,000 First differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Second differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Neither 
D) 4,16,256,65,536,4,294,967,296,18,446,744,073,709,600,000 First differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Second differences: 228,65,040,4,294,836,480, $2 \times 10 ^ { 19 }$
Neither 
E) 0,-4,16,-256,65,536,-4,294,967,296 First differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Second differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Quadratic 
A) 0,4,-16,256,-65,536,4,294,967,296 First differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Second differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Linear 
B) 0,4,16,256,65,536,4,294,967,296 First differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Second differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Quadratic 
C) 4,-16,256,-65,536,4,294,967,296,-18,446,744,073,709,600,000 First differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Second differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Neither 
D) 4,16,256,65,536,4,294,967,296,18,446,744,073,709,600,000 First differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Second differences: 228,65,040,4,294,836,480, $2 \times 10 ^ { 19 }$
Neither 
E) 0,-4,16,-256,65,536,-4,294,967,296 First differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Second differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Quadratic

Question 12

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​
A1 = 2
An = an - 1 + 2
​

Accepted Answer

A) 2,4,6,8,10,12 First differences: 0,0,0,0
Second differences: 2,2,2,2,2
Quadratic 
B) 0,2,4,6,8,10 First differences: 0,0,0,0
Second differences: 2,2,2,2,2
Linear 
C) 0,2,4,6,8,10 First differences: 2,2,2,2,2
Second differences: 0,0,0,0
Quadratic 
D) 2,4,6,8,10,12 First differences: 2,2,2,2,2
Second differences: 0,0,0,0
Linear 
E) 0,2,4,6,8,10 First differences: 2,2,2,2,2
Second differences: -1,-1,-1,-1
Quadratic 
A) 2,4,6,8,10,12 First differences: 0,0,0,0
Second differences: 2,2,2,2,2
Quadratic 
B) 0,2,4,6,8,10 First differences: 0,0,0,0
Second differences: 2,2,2,2,2
Linear 
C) 0,2,4,6,8,10 First differences: 2,2,2,2,2
Second differences: 0,0,0,0
Quadratic 
D) 2,4,6,8,10,12 First differences: 2,2,2,2,2
Second differences: 0,0,0,0
Linear 
E) 0,2,4,6,8,10 First differences: 2,2,2,2,2
Second differences: -1,-1,-1,-1
Quadratic

Question 13

The table shows the numbers $a _ { n }$ (in thousands)of residents from 2002 through 2007. $$\begin{array} { | c | l | } 
\hline  & \text { Number of residents, } \
&.\
{ \text { Year } }& a _ { n } \
&.\
\hline 2002 & 640 \
\hline 2003 & 655 \
\hline 2004 & 670 \
\hline
\end{array}$$ $$\begin{array} { | l | l | } 
\hline 2005 & 670 \
\hline 2006 & 683 \
\hline 2007 & 699 \
\hline
\end{array}$$
​
Find the first differences of the data shown in the table.

Accepted Answer

A)  $15,15,0,13,16$ 
B)  $15,13,0,15,16$ 
C)  $16,0,15,13,15$ 
D)  $0,15,0,13,16$ 
E)  $13,13,15,0,16$ 
A)  $15,15,0,13,16$ 
B)  $15,13,0,15,16$ 
C)  $16,0,15,13,15$ 
D)  $0,15,0,13,16$ 
E)  $13,13,15,0,16$

Question 14

Use mathematical induction to solve for all positive integers n.​ $16 + 34 + 52 + 70 + \ldots + ( 18 n - 2 ) = ?$ ​

Accepted Answer

A)  $\frac { n ( n + 16 ) ( 16 n + 1 ) } { 6 }$ 
B)  $\frac { n } { 2 } ( 3 n + 1 )$ 
C)  $\frac { n } { 2 } ( 3 n + 14 )$ 
D)  $n ( n + 16 )$ 
E)  $\frac { n ( n + 16 ) } { 2 }$ 
A)  $\frac { n ( n + 16 ) ( 16 n + 1 ) } { 6 }$ 
B)  $\frac { n } { 2 } ( 3 n + 1 )$ 
C)  $\frac { n } { 2 } ( 3 n + 14 )$ 
D)  $n ( n + 16 )$ 
E)  $\frac { n ( n + 16 ) } { 2 }$

Question 15

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = 5 , a _ { 2 } = 5 , a _ { 5 } = 65$

Accepted Answer

A)  $a _ { n } = 4 n ^ { 2 } + 5$ 
B)  $a _ { n } = 8 n + 5$ 
C)  $a _ { n } = 4 n ^ { 2 } - 8 n - 5$ 
D)  $a _ { n } = 4 n ^ { 2 } - 8 n + 5$ 
E)  $a _ { n } = 4 n ^ { 2 } - 8 n + 5$ 
A)  $a _ { n } = 4 n ^ { 2 } + 5$ 
B)  $a _ { n } = 8 n + 5$ 
C)  $a _ { n } = 4 n ^ { 2 } - 8 n - 5$ 
D)  $a _ { n } = 4 n ^ { 2 } - 8 n + 5$ 
E)  $a _ { n } = 4 n ^ { 2 } - 8 n + 5$

Question 16

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 9 } n ^ { 3 }$

Accepted Answer

A) 729 
B) 285 
C) 4050 
D) 1296 
E) 2025 
A) 729 
B) 285 
C) 4050 
D) 1296 
E) 2025

Question 17

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​
A1 = 0
An = an - 1 + 5
​

Accepted Answer

A) 0,5,10,15,20,25 First differences: 0,0,0,0
Second differences: 5,5,5,5,5
Quadratic 
B) 0,5,10,15,20,25 First differences: 5,5,5,5,5
Second differences: 0,0,0,0
Linear 
C) 0,5,10,15,20,25 First differences: 5,5,5,5,5
Second differences: -1,-1,-1,-1
Quadratic 
D) 0,5,10,15,20,25 First differences: 5,5,5,5,5
Second differences: 0,0,0,0
Quadratic 
E) 0,5,10,15,20,25 First differences: 0,0,0,0
Second differences: 5,5,5,5,5
Linear 
A) 0,5,10,15,20,25 First differences: 0,0,0,0
Second differences: 5,5,5,5,5
Quadratic 
B) 0,5,10,15,20,25 First differences: 5,5,5,5,5
Second differences: 0,0,0,0
Linear 
C) 0,5,10,15,20,25 First differences: 5,5,5,5,5
Second differences: -1,-1,-1,-1
Quadratic 
D) 0,5,10,15,20,25 First differences: 5,5,5,5,5
Second differences: 0,0,0,0
Quadratic 
E) 0,5,10,15,20,25 First differences: 0,0,0,0
Second differences: 5,5,5,5,5
Linear

Question 18

Use mathematical induction to solve for all positive integers n. ​
A factor of $\left( n ^ { 3 } + 7 n ^ { 2 } + 6 n \right)$ is:
​

Accepted Answer

A) 7 
B) 9 
C) 8 
D) 6 
E) 3 
A) 7 
B) 9 
C) 8 
D) 6 
E) 3

Question 19

Use mathematical induction to solve for all positive integers n.​ $22 + 27 + 32 + 37 + \ldots + ( 5 n - 17 ) = ?$ ​

Accepted Answer

A)  $\frac { n } { 2 } ( 5 n + 5 )$ 
B)  $n ( 22 n + 1 )$ 
C)  $\frac { n ( n + 1 ) } { 22 }$ 
D)  $n ( n + 22 )$ 
E)  $\frac { n } { 2 } ( 5 n + 1 )$ 
A)  $\frac { n } { 2 } ( 5 n + 5 )$ 
B)  $n ( 22 n + 1 )$ 
C)  $\frac { n ( n + 1 ) } { 22 }$ 
D)  $n ( n + 22 )$ 
E)  $\frac { n } { 2 } ( 5 n + 1 )$

Question 20

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { i = 1 } ^ { 7 } \left( 5 i - 8 i ^ { 3 } \right)$$ ​

Accepted Answer

A) -784 
B) -3,136 
C) 4,200 
D) 5,600 
E) -6,132 
A) -784 
B) -3,136 
C) 4,200 
D) 5,600 
E) -6,132

Exam 55: Mathematical Induction

Find the sum using the formulas for the sums of powers of integers.​ ∑n=116n\sum _ { n = 1 } ^ { 16 } n∑n=116​n ​

Determine whether the statement is true or false. ​A sequence with terms has n-1 second differences. ​

Find the sum using the formulas for the sums of powers of integers.​ ∑n=12n4\sum _ { n = 1 } ^ { 2 } n ^ { 4 }∑n=12​n4 ​

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​ A0 = 1 An = an - 1 + n ​

Find the sum using the formulas for the sums of powers of integers. ∑n=1129n−3n2\sum _ { n = 1 } ^ { 12 } 9 n - 3 n ^ { 2 }∑n=112​9n−3n2

Find the sum using the formulas for the sums of powers of integers.​ ∑n=16n2\sum _ { n = 1 } ^ { 6 } n ^ { 2 }∑n=16​n2 ​

Use mathematical induction to solve for all positive integers n.​ 3+8+13+18+…+(5n−2)=?3 + 8 + 13 + 18 + \ldots + ( 5 n - 2 ) = ?3+8+13+18+…+(5n−2)=? ​

Find pk+1 for the given pk.? pk=k27(k+2)2p _ { k } = \frac { k ^ { 2 } } { 7 ( k + 2 ) ^ { 2 } }pk​=7(k+2)2k2​ ?

Find a formula for the sum of the n terms of the sequence. 12,54,258,12516,…\frac { 1 } { 2 } , \frac { 5 } { 4 } , \frac { 25 } { 8 } , \frac { 125 } { 16 } , \ldots21​,45​,825​,16125​,…

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither.​ =4 = ​

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​ A1 = 2 An = an - 1 + 2 ​

The table shows the numbers ana _ { n }an​ (in thousands)of residents from 2002 through 2007. Number of residents, . Year . 2002 640 2003 655 2004 670 2005 670 2006 683 2007 699 ​ Find the first differences of the data shown in the table.

Use mathematical induction to solve for all positive integers n.​ 16+34+52+70+…+(18n−2)=?16 + 34 + 52 + 70 + \ldots + ( 18 n - 2 ) = ?16+34+52+70+…+(18n−2)=? ​

Find a quadratic model for the sequence with the indicated terms. a0=5,a2=5,a5=65a _ { 0 } = 5 , a _ { 2 } = 5 , a _ { 5 } = 65a0​=5,a2​=5,a5​=65

Find the sum using the formulas for the sums of powers of integers. ∑n=19n3\sum _ { n = 1 } ^ { 9 } n ^ { 3 }∑n=19​n3

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​ A1 = 0 An = an - 1 + 5 ​

Use mathematical induction to solve for all positive integers n. ​ A factor of (n3+7n2+6n)\left( n ^ { 3 } + 7 n ^ { 2 } + 6 n \right)(n3+7n2+6n) is: ​

Use mathematical induction to solve for all positive integers n.​ 22+27+32+37+…+(5n−17)=?22 + 27 + 32 + 37 + \ldots + ( 5 n - 17 ) = ?22+27+32+37+…+(5n−17)=? ​

Find the sum using the formulas for the sums of powers of integers.​ ∑i=17(5i−8i3)\sum _ { i = 1 } ^ { 7 } \left( 5 i - 8 i ^ { 3 } \right)∑i=17​(5i−8i3) ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 16 } n$

Determine whether the statement is true or false. A sequence with terms has n-1 second differences.

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 2 } n ^ { 4 }$

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₀ = 1 A_n = an_{- 1} + n

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 12 } 9 n - 3 n ^ { 2 }$

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 6 } n ^ { 2 }$

Use mathematical induction to solve for all positive integers n. $3 + 8 + 13 + 18 + \ldots + ( 5 n - 2 ) = ?$

Find p_k+1 for the given pk.? $p _ { k } = \frac { k ^ { 2 } } { 7 ( k + 2 ) ^ { 2 } }$ ?

Find a formula for the sum of the n terms of the sequence. $\frac { 1 } { 2 } , \frac { 5 } { 4 } , \frac { 25 } { 8 } , \frac { 125 } { 16 } , \ldots$

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. =4 =

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 2 A_n = an_{- 1} + 2

The table shows the numbers $a _ { n }$ (in thousands)of residents from 2002 through 2007. Number of residents, . Year . 2002 640 2003 655 2004 670 2005 670 2006 683 2007 699 Find the first differences of the data shown in the table.

Use mathematical induction to solve for all positive integers n. $16 + 34 + 52 + 70 + \ldots + ( 18 n - 2 ) = ?$

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = 5 , a _ { 2 } = 5 , a _ { 5 } = 65$

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 9 } n ^ { 3 }$

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 0 A_n = an_{- 1} + 5

Use mathematical induction to solve for all positive integers n. A factor of $\left( n ^ { 3 } + 7 n ^ { 2 } + 6 n \right)$ is:

Use mathematical induction to solve for all positive integers n. $22 + 27 + 32 + 37 + \ldots + ( 5 n - 17 ) = ?$

Find the sum using the formulas for the sums of powers of integers. $\sum _ { i = 1 } ^ { 7 } \left( 5 i - 8 i ^ { 3 } \right)$