Deck 55: Mathematical Induction

Question 1

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { n = 1 } ^ { 7 } n ^ { 5 }$$ ​

A)840
B)29,008
C)4,676
D)784
E)140

Answer 1

29,008

Question 2

Find a quadratic model for the sequence with the indicated terms. ​
A₀ = 3,a₁ = 3,a₄ = 15
​

A)an = n² - n + 15
B)an = n² + n - 3
C)an = n² - n - 3
D)an = n² + n + ​3
E)an = n² - n + 3

Answer 2

an = n² - n + 3

Question 3

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

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

Answer 3

-6,132

Question 4

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

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

Question 5

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { n = 1 } ^ { 5 } \left( n ^ { 2 } - n \right)$$ ​

A)40
B)330
C)225
D)15
E)55

Question 6

Find pk _{+ 1} for the given pk.​ $$p _ { k } = \frac { 5 } { ( k + 6 ) ( k + 5 ) }$$ ​

A) $$p _ { k + 1 } = \frac { 7 } { ( k + 6 ) ( k + 6 ) }$$
B) $p _ { k + 1 } = \frac { 5 } { ( k + 7 ) ( k + 7 ) }$
C) $p _ { k + 1 } = \frac { 5 } { ( k + 6 ) ( k + 6 ) }$
D) $$p _ { k + 1 } = \frac { 5 } { ( k + 7 ) ( k + 6 ) }$$
E) $p _ { k + 1 } = \frac { 6 } { ( k + 7 ) ( k + 6 ) }$

Question 7

Find pk _{+ 1} for the given pk .​ $$p _ { k } = \frac { k ^ { 2 } ( k + 4 ) ^ { 2 } } { 8 }$$ ​

A) $p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } ( k + 5 ) ^ { 2 } } { 8 }$
B) $p _ { k + 1 } = \frac { k ^ { 2 } ( k + 5 ) ^ { 2 } } { 8 }$
C) $p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } ( k + 5 ) ^ { 2 } } { 9 }$
D) $p _ { k + 1 } = \frac { ( k + 1 ) ^ { 2 } ( k + 9 ) ^ { 2 } } { 8 }$
E) $p _ { k + 1 } = \frac { k ^ { 2 } ( k + 9 ) ^ { 2 } } { 8 }$

Question 8

Find a quadratic model for the sequence with the indicated terms. ​
A₀ = 8,a₁ = 4,a₃ = 10
​

A)a_n = $\frac { 7 } { 3 }$ n² $- \frac { 19 } { 3 }$ n - 8
B)a_n = 8n² $- \frac { 19 } { 3 }$ n + $\frac { 7 } { 3 }$
C)a_n = $- \frac { 19 } { 3 }$ n² + $\frac { 7 } { 3 }$ n - 8
D)a_n = 8n² $- \frac { 19 } { 3 }$ n - $\frac { 7 } { 3 }$
E)a_n = $\frac { 7 } { 3 }$ n² $- \frac { 19 } { 3 }$ n + 8

Question 9

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { n = 1 } ^ { 22 } \left( n ^ { 3 } - n \right)$$ ​

A)22,770
B)3,795
C)63,756
D)256,036
E)64,009

Question 10

Find pk _{+ 1} for the given p_k.​ $$p _ { k } = \frac { k } { 5 } ( 8 k + 1 )$$ ​

A) $p _ { k + 1 } = \frac { k + 1 } { 5 } ( 8 k + 5 )$
B) $p _ { k + 1 } = \frac { k + 1 } { 5 } ( 8 k + 6 )$
C) $p _ { k + 1 } = \frac { k + 1 } { 6 } ( 8 k + 6 )$
D) $p _ { k + 1 } = \frac { k + 1 } { 5 } ( 8 k + 9 )$
E) $p _ { k + 1 } = \frac { k + 1 } { 6 } ( 8 k + 9 )$

Question 11

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

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 12

Find a quadratic model for the sequence with the indicated terms. ​
A₀ = 4,a₂ = 0,a₆ = 38
​

A)an = -4n² $- \frac { 35 } { 6 }$ n + $\frac { 23 } { 12 }$
B)a_n = $\frac { 23 } { 12 }$ n² $- \frac { 35 } { 6 }$ n - 4
C)​an = 4n² $- \frac { 35 } { 6 }$ n + $\frac { 23 } { 12 }$
D)​an = $- \frac { 35 } { 6 }$ n² + $\frac { 23 } { 12 }$ n - 4
E)​an = $\frac { 23 } { 12 }$ n² $- \frac { 35 } { 6 }$ n + 4

Question 13

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { j = 1 } ^ { 11 } \left( 4 - \frac { 1 } { 2 } j + \frac { 1 } { 2 } j ^ { 2 } \right)$$ ​

A)-264
B)264
C)-506
D)759
E)506

Question 14

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

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

Question 15

Find pk _{+ 1} for the given pk.​ $p _ { k } = \frac { 1 } { 4 ( k + 2 ) }$ ​

A) $p _ { k + 1 } = \frac { 1 } { k ( k + 3 ) }$
B) $p _ { k + 1 } = \frac { 4 } { ( k + 1 ) ( k + 2 ) }$
C) $p _ { k + 1 } = \frac { 4 } { k ( k + 3 ) }$
D) $p _ { k + 1 } = \frac { 1 } { ( k + 3 ) ( k + 2 ) }$
E) $p _ { k + 1 } = \frac { 1 } { 4 ( k + 3 ) }$

Question 16

Find a quadratic model for the sequence with the indicated terms. ​
A₀ = -3,a₂ = 2,a₄ = 10
​

A)an = $\frac {3 } { 8 }$ n² - $\frac { 7 } { 4 }$ n - 3
B)a_n = 3n² + $\frac { 7 } { 4 }$ n + $\frac {3 } { 8 }$
C)a_n = $\frac {3 } { 8 }$ n² + $\frac { 7 } { 4 }$ n + 3
D)a_n = $\frac {3 } { 8 }$ n² + $\frac { 7 } { 4 }$ n - 3
E)an = -3n²+ $\frac { 7 } { 4 }$ n + $\frac {3 } { 8 }$

Question 17

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { n = 1 } ^ { 14 } n$$ ​

A)11,025
B)210
C)6,090
D)105
E)1,015

Question 18

Find the sum using the formulas for the sums of powers of integers.​ $$\sum _ { n = 1 } ^ { 14 } n ^ { 3 }$$ ​

A)6,090
B)105
C)11,025
D)1,015
E)210

Question 19

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

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

Question 20

Find pk _{+ 1} for the given pk.​ $$p _ { k } = \frac { 6 } { k ( k + 1 ) }$$ ​

A) $p _ { k + 1 } = \frac { 2 } { k ( k + 2 ) }$
B) $p _ { k + 1 } = \frac { 6 } { ( k + 1 ) ( k + 1 ) }$
C) $p _ { k + 1 } = \frac { 6 } { ( k + 1 ) ( k + 2 ) }$
D) $p _ { k + 1 } = \frac { 6 } { ( k + 2 ) ( k + 6 ) }$
E) $p _ { k + 1 } = \frac { 6 } { k ( k + 2 ) }$

Question 21

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

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 22

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. ​
A​₁ = 3
A_n = n - an _{- 1}
​

A)3,-1,4,0,5,1 First differences: -9,9,-9,9
Second differences: -4,5,-4,5,-4
Linear
B)0,3,-1,4,0,5 First differences: -4,5,-4,5,-4
Second differences: -9,-9,-9,-9
Quadratic
C)3,-1,4,0,5,1 First differences: -9,-9,-9,-9
Second differences: -4,5,-4,5,-4
Neither
D)3,-1,4,-0,5,-1 First differences: -9,-9,-9,-9
Second differences: -4,5,-4,5,-4
Linear
E)3,-1,4,0,5,1 First differences: -4,5,-4,5,-4
Second differences: 9,-9,9,-9
Neither

Question 23

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 = 1
A_n = an _{- 1} + 2n
​

A)1,5,11,19,29,41 First differences: 4,6,8,10,12
Second differences: 2,2,2,2
Quadratic
B)1,5,11,19,29,41 First differences: 4,6,8,10,12
Second differences: -2,-2,-2,-2
Quadratic
C)1,5,11,19,29,41 First differences: 2,2,2,2
Second differences: 4,6,8,10,12
Linear
D)1,5,11,19,29,41 First differences: -2,-2,-2,2
Second differences: 4,6,8,10,12
Neither
E)1,5,11,19,29,41 First differences: -2,2,-2,2
Second differences: 4,6,8,10,12
Linear

Question 24

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
​

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 25

Use mathematical induction to solve for all positive integers n.​ $4 + 8 + 12 + 16 + \ldots + 2 n = ?$ ​

A) $( n + 3 ) n$
B) $\frac { n ( n + 2 ) } { 2 }$
C) $( n + 2 ) n$
D) $\left( \frac { n ( n + 2 ) } { 2 } \right) ^ { 2 }$
E) $\frac { n ( n + 1 ) ( n + 2 ) } { 6 }$

Question 26

Find a quadratic model for the sequence with the indicated terms.​ $a _ { 0 } = - 3 , a _ { 2 } = - 7 , a _ { 6 } = - 57$ ​

A)an = $\frac { 3 } { 2 }$ n² - $\frac { 7 } { 4 }$ n - 3
B)an = -3n² + $\frac { 3 } { 2 }$ n - $\frac { 7 } { 4 }$
C)an = $- \frac { 7 } { 4 }$ n² + $\frac { 3 } { 2 }$ n - 3
D)an = - $\frac { 7 } { 4 }$ n² + $\frac { 3 } { 2 }$ n + 3
E)an = 3n² + $\frac { 3 } { 2 }$ n - $\frac { 7 } { 4 }$

Question 27

Use mathematical induction to solve for all positive integers n.​ $1 + 2 + 3 + 4 + \ldots + n = ?$ ​

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

Question 28

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. ​
A₂ = -4
A_n = -2an _{- 1}
​

A)-4,8,-16,32,-64,128 First differences: 12,-24,48,-96,192
Second differences: -36,72,-144,288
Neither
B)0,4,8,16,32,64 First differences: 12,24,48,96,192
Second differences: -36,72,-144,288
Quadratic
C)0,4,-8,16,-32,64 First differences: -36,72,-144,288
Second differences: 12,24,48,96,192
Linear
D)0,-4,8,-16,32,-64 First differences: 12,24,48,96,192
Second differences: -36,72,-144,288
Quadratic
E)4,-8,16,-32,64,-128 First differences: -36,72,-144,288
Second differences: 12,24,48,96,192
Neither

Question 29

Find a formula for the sum of the first n terms of the sequence.​ $9,12,15,18,21 , \ldots$ ​

A) $S _ { n } = \frac { n } { 2 } ( 3 n + 15 )$
B) $S _ { n } = \frac { n } { 2 } ( 3 n + 3 )$
C) $S _ { n } = \frac { n } { 2 } ( - 3 n - 15 )$
D) $S _ { n } = \frac { n } { 2 } ( 3 n - 15 )$
E) $S _ { n } = \frac { n } { 2 } ( - 3 n + 15 )$

Question 30

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 = 3
​a_n = an _{- 1} - n

A)0,3,1,-2,-6,-11 First differences: 3,-2,-3,-4,-5
Second differences: -1,-1,-1,-1
Quadratic
B)0,3,1,-2,-6,-11,-17 First differences: -2,-3,-4,-5,-6
Second differences: 1,1,1,1
Linear
C)3,1,-2,-6,-11,-17 First differences: -2,-3,-4,-5,-6
Second differences: 1,1,1,1
Quadratic
D)3,1,-2,-6,-11,-17 First differences: -2,-3,-4,-5,-6
Second differences: -1,-1,-1,-1
Quadratic
E)0,3,1,-2,-6,-11 First differences: 3,-2,-3,-4,-5
Second differences: 1,1,1,1
Quadratic

Question 31

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.

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 32

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

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

Question 33

Determine whether the statement is true or false. ​If the statement P_1​ is true but the true statement P₆ does not imply that the statement P₇ is true,then P_n is not
Necessarily true for all positive integers n.
​
​

A)False
B)True

Question 34

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

A)False
B)True

Question 35

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

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 36

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
​

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 37

Use mathematical induction to solve for all positive integers n.​ $5 + 9 + 13 + 17 + \ldots + ( 4 n + 1 ) = ?$ ​

A) $n ( 9 n + 3 )$
B) $\frac { n ( 2 n + 3 ) } { 2 }$
C) $n ( n + 3 )$
D) $\frac { n ( n + 3 ) } { 9 }$
E) $n ( 2 n + 3 )$

Question 38

Find a formula for the sum of the first n terms of the sequence.​ $17,21,25,29,33 , \ldots$ ​

A) $S _ { n } = n ( 2 n + 15 )$
B) $S _ { n } = n ( 5 n + 1 )$
C) $S _ { n } = \frac { ( n + 1 ) } { n }$
D) $S _ { n } = n ( 2 n + 1 )$
E) $S _ { n } = n ( 17 n - 1 )$

Question 39

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
​

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 40

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.​ $\begin{array} { l } a _ { 0 } = 4 \\a _ { n } = \left( a _ { n - 1 } \right) ^ { 2 }\end{array}$ ​

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 41

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.

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 42

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

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

Question 43

Find Pk₊₁ for the given P_k. $$P _ { k } = \frac { 2 } { k ( k + 1 ) }$$

A) $$P _ { k + 1 } = \frac { 2 } { k ( k + 2 ) }$$
B) $$P _ { k + 1 } = \frac { 2 } { k ( k + 1 ) } + \frac { 2 } { ( k + 1 ) ( k + 2 ) }$$
C) $P _ { k + 1 } = \frac { 4 } { ( k + 1 ) ( k + 2 ) }$
D) $P _ { k + 1 } = \frac { 2 } { ( k + 1 ) ( k + 2 ) }$
E) $P _ { k + 1 } = \frac { 2 } { k ( k + 1 ) } + 1$

Question 44

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

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 45

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

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 46

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

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

Question 47

Use mathematical induction to solve for all positive integers n.​ $\sum _ { i = 1 } ^ { n } i ( i + 4 ) = ?$ ​

A) $\frac { n ( 2 n + 13 ) } { 6 }$
B) $\frac { n ( n + 1 ) ( 2 n + 13 ) } { 4 }$
C) $\frac { n ( n + 1 ) ( n + 13 ) } { 6 }$
D) $\frac { n ( n + 1 ) ( 2 n + 13 ) } { 6 }$
E) $\frac { n ( n + 1 ) ( 2 n + 13 ) } { 2 }$

Question 48

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

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