Deck 55: Mathematical Induction

Question 1

Find the sum using the formulas for the sums of powers of integers.​ ​

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.​ ​

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.​ ​

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.​ ​

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

Question 6

Find pk _{+ 1} for the given pk.​ ​

A)
B)
C)
D)
E)

Question 7

Find pk _{+ 1} for the given pk .​ ​

A)
B)
C)
D)
E)

Question 8

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

A)a_n = n² n - 8
B)a_n = 8n² n +
C)a_n = n² + n - 8
D)a_n = 8n² n -
E)a_n = n² n + 8

Question 9

Find the sum using the formulas for the sums of powers of integers.​ ​

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.​ ​

A)
B)
C)
D)
E)

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 n +
B)a_n = n² n - 4
C)​an = 4n² n +
D)​an = n² + n - 4
E)​an = n² n + 4

Question 13

Find the sum using the formulas for the sums of powers of integers.​ ​

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.​ ​

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

Question 15

Find pk _{+ 1} for the given pk.​ ​

A)
B)
C)
D)
E)

Question 16

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

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

Question 17

Find the sum using the formulas for the sums of powers of integers.​ ​

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.​ ​

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.​ ​

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

Question 20

Find pk _{+ 1} for the given pk.​ ​

A)
B)
C)
D)
E)

Question 21

Use mathematical induction to solve for all positive integers n.​ ​

A)
B)
C)
D)
E)

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.​ ​

A)
B)
C)
D)
E)

Question 26

Find a quadratic model for the sequence with the indicated terms.​ ​

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

Question 27

Use mathematical induction to solve for all positive integers n.​ ​

A)
B)
C)
D)
E)

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.​ ​

A)
B)
C)
D)
E)

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 (in thousands)of residents from 2002 through 2007.
​
Find the first differences of the data shown in the table.

A)
B)
C)
D)
E)

Question 32

Use mathematical induction to solve for all positive integers n. ​
A factor of 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.​ ​

A)
B)
C)
D)
E)

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.​ ​

A)
B)
C)
D)
E)

Question 38

Find a formula for the sum of the first n terms of the sequence.​ ​

A)
B)
C)
D)
E)

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.​ ​

A)0,4,-16,256,-65,536,4,294,967,296 First differences: -228,65,040,-4,294,836,480,
Second differences: 12,240,65,280,4,294,901,760,
Linear
B)0,4,16,256,65,536,4,294,967,296 First differences: 12,240,65,280,4,294,901,760,
Second differences: -228,65,040,-4,294,836,480,
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,
Second differences: 12,240,65,280,4,294,901,760,
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,
Second differences: 228,65,040,4,294,836,480,
Neither
E)0,-4,16,-256,65,536,-4,294,967,296 First differences: 12,240,65,280,4,294,901,760,
Second differences: -228,65,040,-4,294,836,480,
Quadratic

Question 41

The table shows the numbers (in thousands)of residents from 2002 through 2007.
Determine whether a linear model can be used to approximate the data.
If so,find a model algebraically.Let n represent the year,with corresponding to 2002.

A)A linear model can be used.
B)A linear model can be used.
C)A linear model can be used.
D)A linear model can be used.
E)A linear model can be used.

Question 42

Find the sum using the formulas for the sums of powers of integers.

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

Question 43

Find Pk₊₁ for the given P_k.

A)
B)
C)
D)
E)

Question 44

Use mathematical induction to solve for all positive integers n.​ ​

A)
B)
C)
D)
E)

Question 45

Find a formula for the sum of the n terms of the sequence.

A)
B)​
C)
D)
E)

Question 46

Find a quadratic model for the sequence with the indicated terms.

A)
B)
C)
D)
E)

Question 47

Use mathematical induction to solve for all positive integers n.​ ​

A)
B)
C)
D)
E)

Question 48

Find the sum using the formulas for the sums of powers of integers.

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