Free Printable Mathematical Induction

1.

What is the third step in Mathematical induction?

a)

P(1)

b)

P(k+1)

c)

P(k)

d)

n=k

2.

1 + 3 + 5 + 7 + . . . + (2n − 1) = n²

On the basis of this assumption, [The statement is true for n = k:

1 + 3 + 5 + 7 + . . . + (2k − 1) = k²]

What must we show next?

a)

The statement is true for n = 1:

(2)(1) − 1 = 1²

b)

The statement is true for n = k:

1 + 3 + 5 + 7 + . . . + (2k − 1) = k²

c)

The statement is true for n = k + 1:

1 + 3 + 5 + 7 + . . . + (2k − 1) + (2k + 1) = (k + 1)²

3.

Let P(n) = 2n − 1. Evaluate:

a) P(k)

b) P(k + 1)

a)

a) P(k) = 2k − 1

b) P(k + 1) = 2n + 1

b)

a) P(k) = 2k + 1

b) P(k + 1) = 2(k + 1) - 1

c)

a) P(k) = 2k − 1

b) P(k + 1) = 2k + 1

4.

1 + 3 + 5 + 7 + . . . + (2n − 1) = n²

To prove this by mathematical induction, what will be the induction assumption?

a)

The statement is true for n = k:

1 + 3 + 5 + 7 + . . . + (2k − 1) = k²

b)

The statement is true for n = 1:

(2)(1) − 1 = 1²

c)

The statement is true for n = k + 1:

1 + 3 + 5 + 7 + . . . + (2k − 1) + (2k + 1) = (k + 1)²

5.

What is the first step in Mathematical Induction?

a)

P(k)

b)

n=k

c)

P(k+1)

d)

P(1)

6.

According to the principle of mathematical induction, to prove a statement that is asserted about every natural number n, there are two things to prove. What is the first?

a)

The statement is true for n = 1.

b)

The statement is true for n = k.

c)

The statement is true for n = k+1.

7.

According to the principle of mathematical induction, to prove a statement that is asserted about every natural number n, there are two things to prove. What is the second?

a)

The statement is true for n = k+1.

b)

If the statement is true for n = k, then it will be true for its successor, k + 1.

c)

The statement is true for n = 1.

d)

The statement is true for n = k.

8.

1 + 3 + 5 + 7 + . . . + (2n − 1) = n²
On the basis of this assumption,
[The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k²]
What must we show?

a)

The statement is true for n = 1:
2x1 − 1 = 1²

b)

The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k²

c)

The statement is true for n = k + 1:
1 + 3 + 5 + 7 + . . . + (2k − 1) + (2k + 1) = (k + 1)²

9.

Let S(n) = 2n − 1. Evaluate:
a) S(k)
b) S(k + 1)

a)

a) S(k) = 2k − 1
b) S(k + 1) = 2n + 1

b)

a) S(k) = 2k + 1
b) S(k + 1) = 2k + 1

c)

a) S(k) = 2k − 1
b) S(k + 1) = 2k + 1

10.

1 + 3 + 5 + 7 + . . . + (2n − 1) = n²

On the basis of this assumption,

[The statement is true for n = k:

1 + 3 + 5 + 7 + . . . + (2k − 1) = k²]

What must we show next?

a)

The statement is true for n = 1:

2x1 − 1 = 1²

b)

The statement is true for n = k:

1 + 3 + 5 + 7 + . . . + (2k − 1) = k²

c)

The statement is true for n = k + 1:

1 + 3 + 5 + 7 + . . . + (2k − 1) + (2k + 1) = (k + 1)²

11.

When using mathematical induction to prove : $\sum_{i=1}^ni^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$ . In step #2, after you have made your assumption, what are you trying to prove? (What is your goal?)

a)

$\sum_{i=1}^{k+1}i^2=\frac{\left(k\right)\left(k+1\right)\left(2k+1\right)}{6}+\left(k+1\right)^2$

b)

$S_{k+1}=\frac{k\left(k+1\right)\left(2k+1\right)}{6}$

c)

$\sum_{i=1}^{k+1}i^2=\frac{\left(k+1\right)\left(k+2\right)\left(2k+3\right)}{6}$

d)

$S_{k+1}=\left(k+1\right)^2$

12.

What does the principle of mathematical induciton state

a)

if the base case (for n = 1) is true and inductive step is true, then the theorem holds for all positive integers

b)

if the base case (for n = 1) is true and inductive step is false, then the theorem holds for all positive integers

c)

if the base case (for n = 1) is false and inductive step is true, then the theorem holds for all positive integers

d)

if the base case (for n = 1) is false and inductive step is false, then the theorem holds for all positive integers

13.

What value of n will satisfy the base case?

a)

0

b)

1

c)

2

d)

3

14.

Find the coefficient of x³ in...

(2 + x)⁵

a)

40

b)

10

c)

20

d)

80

15.

Find the coefficient of x⁴ in...

(3 + x)⁶

a)

135

b)

15

c)

45

d)

540

16.

Find the coefficient of x³ in...

(1 - x)⁷

a)

-35

b)

35

c)

-21

d)

21

17.

Find the coefficient of x² in...

(2 - x)⁶

a)

240

b)

-240

c)

30

d)

-30

18.

Find the coefficient of x³ in...

(1 + 3x)⁵

a)

270

b)

90

c)

30

d)

810

19.

Find the coefficient of x² in...

(1 + 2x)⁶

a)

60

b)

30

c)

15

d)

120

20.

Find the coefficient of x³ in...

(1 - 5x)⁴

a)

-500

b)

500

c)

100

d)

-100

21.

Find the coefficient of x² in...

(3 - 2x)⁴

a)

216

b)

-219

c)

432

d)

-432

22.

Evaluate $^{\frac{1}{2}}C_1$

a)

$\frac{1}{2}$

b)

$1$

c)

$2$

d)

$\frac{2}{3}$

23.

Choose the right Pascal's triangle

a)

b)

c)

d)

24.

Consider the binomial expansion of $\left(2y-5x\right)^6$ . How many terms are there in the expansion?

a)

6

b)

5

c)

7

d)

4

25.

Use the binomial expansion to expand $\left(u^2-2\right)^3$

a)

$u^6-6u^4+12u^2-8$

b)

$u^6-16u^4+12u^2-8$

c)

$5u^6-20u^4+40u^2-40$

d)

$u^6-24u^4+12u^2-8$

26.

Use the binomial theorem to expand $\left(4-2z\right)^3$

a)

$64-96z+48z^2+8z^3$

b)

$64-96z+48z^2-8z^3$

c)

$64+96z+48z^2+8z^3$

d)

$64-64z+144z^2-8z^3$

27.

What is the first term of $\left(2r-s\right)^8$

a)

$64r$

b)

$64r^8$

c)

$256r^8$

d)

$128r^8$

28.

Using $\left(2y-5x\right)^6$ , find the third term of the expansion.

a)

$6000x^4y^2$

b)

$6000x^3y^5$

c)

$6000x^2y^4$

d)

$6001x^2y^4$

29.

From $\left(x^2-2y\right)^7$ , find the fifth term.

a)

$566x^6y^4$

b)

$550x^6y^5$

c)

$560x^6y^4$

d)

$560x^4y^6$

30.

Consider the binomial expansion of $\left(3x^2-2y^3\right)^5$ . Find the third term of the expansion.

a)

$1080x^6y^6$

b)

$1180x^4y^5$

c)

$1088x^6y^4$

d)

$1011x^6y^6$

31.

Given $\left(3x-\frac{2}{3}y^2\right)^5$ . Use Binomial Theorem to find the fourth term of the expansion.

a)

$-\frac{80}{3}x^2y^6$

b)

$-\frac{80}{3}x^2y^4$

c)

$-\frac{80}{3}x^2y^2$

d)

$-\frac{88}{3}x^2y^6$

32.

Find the coefficient of the term $x^8y^3$ of $\left(3x^2-2y^3\right)^5$

a)

-811

b)

-800

c)

-810

d)

-910

33.

Find the coefficient of $x^5$ in the expansion of $\left(2-x\right)\left(3x+1\right)^9$

a)

91854

b)

88547

c)

11238

d)

51030

34.

$83\times82\times81\times80\times79$ can also be written as:

a)

$\frac{83!}{78!}$

b)

$\frac{83!}{79!}$

c)

$\frac{78!}{83}$

d)

$8378$

35.

True or False...

40,500 is divisible by 6

a)

True

b)

False

36.

Which of the following is NOT divisible by 4?

a)

1,000

b)

5,740

c)

1,566

d)

2,024

37.

All of these numbers are divisible by 5 except...

a)

1050

b)

155

c)

980

d)

2938

38.

Is 2,385 divisible by 10?

a)

Yes

b)

No

39.

105 is NOT divisible by which of the following numbers?

a)

1

b)

5

c)

3

d)

2

40.

Which of these numbers is divisible by 2 AND 5?

a)

675

b)

674

c)

670

d)

673

41.

55 is divisible by what number?

a)

5

b)

2

c)

10

d)

6

42.

543 is divisible by which of the following? Select all that apply.

a)

5

b)

8

c)

10

d)

3

e)

1

43.

684,933 is divisible by which of the following numbers? (select all that apply)

a)

3

b)

5

c)

7

d)

8

e)

9

44.

What is the divisibility rule for 4?

a)

The last two digits form a number divisible by 4

b)

The last digit is a number divisible by 4

c)

The last digit is even

d)

The last digit is zero

45.

What is the divisibility rule for 5?

a)

It is divisible by 2 and 3

b)

The last digit is a five

c)

The last digit is five or zero

d)

The last digit is zero

46.

Which of the following statements is true of this number?

a)

It is divisible by 3 and 9

b)

It is divisible by 2, 3, and 6

c)

It is divisible by 3

d)

It is prime

47.

Which of the following statements is true of this number?

a)

It is divisible by 2, 6, and 9

b)

It is divisible by 3 and 9

c)

It is divisible by 3

d)

It is prime

48.

What is the divisibility rule of 9?

a)

The sum of the digits is a number divisible by 3

b)

The sum of the digits is a number divisible by 9

c)

The last two digits form a number divisible by 3

d)

The last digit is zero

49.

Which of the following is an even number?

a)

1

b)

9

c)

4

d)

5

50.

Is 1,284,604 divisible by 2?

a)

Yes, because the sum of the digits is 20, which is an even number

b)

No, because 4 is an odd number

c)

Yes, because 4 is an even number

d)

No, because the it is not divisible by 3

51.

2520 is divisible by 6 because:

a)

It is divisible by 2 (even) and 5 (ends in a 0 or 5)

b)

The sum of the digits is 12, a multiple of 6

c)

It's divisible by 2 (even) and 3 (sum of digits is a multiple of 3)

d)

The sum of the digits is even

52.

How do you know if a number is divisible by 12?

a)

The sum of the digits must be a multiple of 12

b)

If it ends in 3, 6, or 9

c)

If it is divisible by 3 and 4

d)

If it is divisible by 2 or 3

53.

All of these numbers are divisible by 5 except for...

a)

75,395

b)

984,940

c)

222,222,235

d)

8,943,892

54.

Is 81 divisible by 3?

a)

Yes, because the sum of the digits is 9, a multiple of 3

b)

No, because 81 is an odd number

c)

Yes, because 81 is an odd number

d)

No, because 81 does not end in 3

55.

Is 816 divisible by 3?

a)

Yes

b)

No

56.

Is 36,928 divisible by 4?

a)

Yes, because the sum of the digits is a multiple of 4

b)

Yes, because it is an even number

c)

Yes because the last two digits, 28, is divisible by 4

d)

Yes, because it is divisible by 2 and 3

57.

Is 19,584 divisible by 9?

a)

Yes

b)

No

58.

Any whole number is divisible by 6 if it is divisible by

a)

2 and 3

b)

2 and 4

c)

9 and 1

d)

5 and 10

59.

Insert the correct digit to make the following number divisible by 3.
60__2

a)

1

b)

2

c)

3

d)

5

60.

54,322 is divisible by 3

a)

True

b)

False

61.

What is the divisibility rule for 1

a)

It has to be an even number

b)

Every number is divisible by 1

c)

Add up all of the digits and if it is divisible by 1 then the whole number is.

62.

Is 61 Prime or Composite?

a)

Prime

b)

Composite

63.

insert the greatest digit to make the following number divisible by 3.

98732_6

a)

9

b)

8

c)

7

d)

1

64.

insert the correct digit to make the following number divisible by 9.

987654_2

a)

6

b)

4

c)

13

d)

1

Mathematical Induction

Total questions: 64

Worksheet time: 53mins

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