Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

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)  $p _ { k + 1 } = \frac { k ^ { 2 } ( k + 9 ) ^ { 2 } } { 8 }$ 
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)  $p _ { k + 1 } = \frac { k ^ { 2 } ( k + 9 ) ^ { 2 } } { 8 }$ A

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

A)  $n ( 9 n + 3 )$ 
B)  $\frac { n ( 2 n + 3 ) } { 2 }$ 
C)  $n ( n + 3 )$ 
D)  $\frac { n ( n + 3 ) } { 9 }$ 
E)  $n ( 2 n + 3 )$ 
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 5

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

Accepted Answer

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)  $S _ { n } = n ( 17 n - 1 )$ 
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 6

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

Accepted Answer

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)  $p _ { k + 1 } = \frac { 6 } { k ( k + 2 ) }$ 
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 7

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
An = -2an - 1
​

Accepted Answer

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 
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 8

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

Accepted Answer

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 - $\frac { 7 } { 4 }$ 
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 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

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 = 1
An = an - 1 + 2n
​

Accepted Answer

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 
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 13

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
An = n - an - 1
​

Accepted Answer

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 
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 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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

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

Accepted Answer

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)  $p _ { k + 1 } = \frac { 6 } { ( k + 7 ) ( k + 6 ) }$ 
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 17

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

Accepted Answer

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)  $p _ { k + 1 } = \frac { k + 1 } { 6 } ( 8 k + 9 )$ 
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 18

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

Accepted Answer

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)  $P _ { k + 1 } = \frac { 2 } { k ( k + 1 ) } + 1$ 
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 19

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
​an = an - 1 - n

Accepted Answer

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 
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 20

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

Accepted Answer

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

Exam 55: Mathematical Induction

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

Find pk + 1 for the given pk .​ pk=k2(k+4)28p _ { k } = \frac { k ^ { 2 } ( k + 4 ) ^ { 2 } } { 8 }pk​=8k2(k+4)2​ ​

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

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

Find a formula for the sum of the first n terms of the sequence.​ 17,21,25,29,33,…17,21,25,29,33 , \ldots17,21,25,29,33,… ​

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

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 An = -2an - 1 ​

Find a quadratic model for the sequence with the indicated terms.​ a0=−3,a2=−7,a6=−57a _ { 0 } = - 3 , a _ { 2 } = - 7 , a _ { 6 } = - 57a0​=−3,a2​=−7,a6​=−57 ​

Find the sum using the formulas for the sums of powers of integers.​ ∑j=111(4−12j+12j2)\sum _ { j = 1 } ^ { 11 } \left( 4 - \frac { 1 } { 2 } j + \frac { 1 } { 2 } j ^ { 2 } \right)∑j=111​(4−21​j+21​j2) ​

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

Find the sum using the formulas for the sums of powers of integers.​ ∑n=15(n2−n)\sum _ { n = 1 } ^ { 5 } \left( n ^ { 2 } - n \right)∑n=15​(n2−n) ​

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 An = an - 1 + 2n ​

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 An = n - an - 1 ​

Find the sum using the formulas for the sums of powers of integers.​ ∑n=114n3\sum _ { n = 1 } ^ { 14 } n ^ { 3 }∑n=114​n3 ​

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

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

Find pk + 1 for the given pk.​ pk=k5(8k+1)p _ { k } = \frac { k } { 5 } ( 8 k + 1 )pk​=5k​(8k+1) ​

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

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 ​an = an - 1 - n

Find the sum using the formulas for the sums of powers of integers.​ ∑n=114n\sum _ { n = 1 } ^ { 14 } n∑n=114​n ​

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

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

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

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

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

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

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

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}

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

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

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

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

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

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 = n - an_{- 1}

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

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

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

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

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

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

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