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Chapter 2 l Skills Practice 295

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Skills Practice Skills Practice for Lesson 2.1

Name _____________________________________________ Date ____________________

Quadratic Formula Deriving the Quadratic Formula

Vocabulary Answer each question.

1. What is the Quadratic Formula used to calculate?

2. What is the formula for the Quadratic Formula?

Problem Set Use the Quadratic Formula to calculate the root(s) of each quadratic equation. Check your answers.

1. x2 � 3x � 10 � 0 2. x2 � 7x � 8 � 0

a � 1, b � 3, c � �10

x � �b � √_________

b2 � 4ac ________________ 2a

� �3 � √

______________ 32 � 4(1)(�10) ____________________

2(1)

� �3 � √___

49 __________ 2

x � �3 � 7 _______ 2 or x � �3 � 7 _______

2

x � 4 __ 2 � 2 x � �10 _____

2 � �5

Check:

22 � 3(2) � 10 � 4 � 6 � 10 � 0

(�5)2 � 3(�5) � 10 � 25 � 15 � 10 � 0

296 Chapter 2 l Skills Practice

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3. 2x2 � 4x � 2 � 0 4. �2x2 � 3x � 1 � 0

Use the Quadratic Formula to calculate the zero(s) of each quadratic function. Check your answers.

5. f(x) � x2 � 25 6. f(x) � 3x2 � 5x

Zeros: (�5, 0), (5, 0)

a � 1, b � 0, c � �25

x � �b � √_________

b2 � 4ac ________________ 2a

� �0 � √

______________ 02 � 4(1)(�25) ____________________

2(1)

� 0 � √____

100 _________ 2

x � 10 ___ 2 or x � �10 _____

2

x � 5 x � �5

Check:

(5)2 � 25 � 25 � 25 � 0

(�5)2 � 25 � 25 � 25 � 0


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7. f(x) � (x � 7)2 � 30 8. f(x) � (x � 4)2 � 8

Use any method to solve each quadratic equation. Check your answers.

9. x2 � 6x � 9 � 0 10. x2 � 6x � 8 � 0

x2 � 6x � 9 � 0

( x � 3)( x � 3) � 0

x � 3

Check:

(3)2 � 6(3) � 9 � 9 � 18 � 9 � 0


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11. (x � 3)2 � 3 � 0 12. (x � 6)2 � 4 � 0

Use any method to calculate the zero(s) of each quadratic function. Check your answers.

Zeros: (10.27, 0), (2.73, 0)

x � �b � √_________

b2 � 4ac ________________ 2a

� �13 � √

________________ 132 � 4(�1)(�28) ________________________

2(�1)

� �13 � √___

57 ___________ �2

� 13 � √___

57 _________ 2

x � 13 � √___

57 _________ 2 or x � 13 � √

___ 57 _________

2

x � 10.27 x � 2.73 Check:

� ( 13 � √___

57 _________ 2

) 2 � 13 ( 13 � √___

57 _________ 2 ) � 28 � 0

� ( 13 � √___

57 _________ 2

) 2 � 13 ( 13 � √___

57 _________ 2 ) � 28 � 0

13. f(x) � �x2 � 13x � 28 14. f(x) � 2x2 � 8x � 42


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15. f( x) � ( x � 4)2 � 9 16. f( x) � ( x � 7)2 � 25


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Calculating Roots and Zeros Solving Quadratic Equations and Inequalities

Vocabulary Explain how each set of terms are related.

1. equation and inequality

2. roots of a quadratic equation and zeros of a quadratic function

Problem Set Use the Quadratic Formula to solve each quadratic equation. Check your solution.

1. x2 � 7x � 18 � 0

a � 1, b � 7, c � �18

x � �b � √_________

b2 � 4ac ________________ 2a

� �7 � √

______________ 72 � 4(1)(�18) ____________________

2(1) � �7 � √

________ 49 � 72 _______________

2 � �7 � √

____ 121 ___________

2

x � �7 � 11 ________ 2

or x � �7 � 11 ________ 2

x � 4 __ 2 � 2 x � �18 _____

2 � �9

Check:

22 � 7(2) � 18 � 4 � 14 � 18 � 0

(�9)2 � 7(�9) � 18 � 81 � 63 � 18 � 0


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2. x2 � 55x � 650 � 0

3. �2x2 � 3x � 7 � 0

4. 4x2 � x � 3 � 0


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Use the Quadratic Formula to calculate the zeros of each quadratic function. Check your solution.

5. f(x) � �3x2 � 4x � 4

Zeros: ( � 2 __ 3

, 0 ) , (2, 0)

a � �3, b � 4, c � 4

x � �b � √_________

b2 � 4ac ________________ 2a

� �4 � √

_____________ 42 � 4(�3)(4) ___________________

2(�3) � �4 � √

________ 16 � 48 _______________

�6 � �4 � √

___ 64 __________

�6 � �4 � 8 _______

�6

x � � 2 __ 3 x � 2

Check:

�3 ( � 2 __ 3 ) 2 � 4 ( � 2 __

3 ) � 4 � � 4 __

3 � 8 __

3 � 4 � � 12 ___

3 � 4 � 0

�3(2)2 � 4(2) � 4 � �12 � 8 � 4 � 0

6. f(x) � 5x2 � 8x � 3

7. f(x) � 2x2 � 5x � 18


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8. f(x) � �3x2 � 7x � 2

Use the given information to answer each question.

9. A ball is tossed upward from a height of 1 meter with an initial velocity of 4 meters

per second. Its height can be modeled with the quadratic function h(t) � �5t2 � 4t � 1,

where t is the time in seconds and h is the height of the ball in meters t seconds after it

is thrown.

a. How long will it take the ball to reach its maximum height?

�b ____ 2a

� �4 ______ 2(�5)

� 4 ___ 10

� 0.4

The ball reaches its maximum height 0.4 second after it is thrown.

b. How long does it take for the ball to reach the ground?

h(t) � �5t2 � 4t � 1 � 0

a � �5, b � 4, c � 1

t � �b � √_________

b2 � 4ac ________________ 2a

� �4 � √

_____________ 42 � 4(�5)(1) ___________________

2(�5) �

�(4) � √________

16 � 20 ________________

�10 � �4 � √

___ 36 __________

�10 � �4 � 6 _______

�10

t � �0.2 or t � 1

The ball reaches the ground after 1 second.


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10. A water balloon is thrown upward from a height of 10 feet with an initial velocity of

46 feet per second. Its height can be modeled with the quadratic function

h(t) � �16t2 � 46t � 10, where t is the time in seconds and h is the height of the

balloon in feet t seconds after it is thrown.

a. What is the maximum height the balloon reaches?

b. How long does it take for the balloon to reach the ground?


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11. A golf ball is thrown upward from a height of 5 feet with an initial velocity of 20 feet per

second. Its height can be modeled by the quadratic function h(t) � �16t2 � 20t � 5, where t

is the time in seconds and h is the height of the golf ball in feet t seconds after it is thrown.

a. How long will it take the golf ball to reach its maximum height?

b. How long will it take for the golf ball to reach the ground?

12. A soccer ball is thrown upward from a height of 4 meters with an initial velocity of 8 meters

per second. Its height can be modeled by the quadratic function h(t) � �5t2 � 8t � 4,

where t is the time in seconds and h is the height of the soccer ball in meters t seconds

after it is thrown.

a. How long will it take the soccer ball to reach its maximum height?

b. How long will it take for the soccer ball to reach the ground?


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Use factoring to solve each quadratic inequality. Check your solution.

13. 2x2 � 5x � 5 � 2

2 x2 � 5x � 5 � 2

2 x2 � 5x � 7 � 0

(2 x � 7)( x � 1) � 0

(2 x � 7) � 0 or ( x � 1) � 0

2 x � �7 x � 1

x � �7 ___ 2

Try: x � �9, x � 0, and x � 2

2(�9)2 � 5(�9) � 5 � 2(81) � 45 � 5 � 112 112 � 2

2(0)2 � 5(0) � 5 � 0 � 0 � 5 � �5 �5 is not � �2

2(2)2 � 5(2) � 5 � 8 � 10 � 5 � 13 13 � 2

Solution: x � ( �∞, � 7 __ 2

] or x � [1, ∞)

14. 2 x2 � 14x � 27 � 7


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15. 4x2 � 15x � 25

16. 3x2 � 10x � �3


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The DiscriminantThe Discriminant and the Nature of Roots/Vertex Form

Vocabulary Define each term or phrase in your own words.

1. number of zeros of a quadratic function

2. nature of the roots/zeros in a quadratic equation

3. discriminant

4. vertex form of a quadratic equation


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Problem Set Use the discriminant to determine the number and nature of the roots of each quadratic equation. Then solve for the root(s), if possible.

1. x2 � 5x � 7 � 0

a � 1, b � �5, c � 7

b2 � 4ac � (�5)2 � 4(1)(7) � 25 � 28 � �3

The equation has no roots because the discriminant is negative.

2. x2 � 8x � 3 � 0

3. 9x2 � 6x � 1 � 0

4. 5x2 � 2x � 1 � 0


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Use the discriminant to determine the number of times the graph of each quadratic function intersects the x-axis. Calculate the x-intercepts.

5. f(x) � �2x2 � 7x � 3

a � �2, b � 7, c � 3

b2 � 4ac � 72 � 4(�2)(3) � 49 � 24 � 73

The graph of the function intercepts the x-axis two times.

x � �b � √_________

b2 � 4ac ________________ 2a

� �7 � √

_____________ 72 � 4(�2)(3) ___________________

2(�2) � �7 � √

___ 73 __________

�4

x � 3.886 or x � �0.386

x-intercepts approximately: (3.886, 0), (�0.386, 0)

6. f( x) � 2x2 � 4x � 2

7. f( x) � �2x2 � 8x � 42

8. f( x) � �5x2 � 10x � 5


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Write each equation in vertex form. Then determine the vertex. Check your solution.

9. y � x2 � 4x � 7 10. y � x2 � 8x � 12

y � ( x2 � 4x � 4) � 7 � 4

y � ( x � 2)2 � 11

Vertex: (2, �11)

Check:

�b ____ 2a

� 4 ____ 2(1)

� 2

y � (2)2 � 4(2) � 7

y � 4 � 8 � 7 � �11

11. y � 3x2 � 18x � 5 12. y � 2x2 � 8x � 9

13. y � x2 � 3x � 28 14. y � x2 � 17x � 74


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The Complete Number SystemOperations with Complex Numbers

Vocabulary Write the term that best completes each statement.

1. Calculating the of a complex number can be accomplished by

repeated multiplication.

2. The of a complex number is used to divide complex numbers.

Problem Set Write the conjugate for each complex number.

1. �6 � 12 i 2. a � bi

�6 � 12 i

3. 2 i 4. �3 � 10i

5. 16 � 12 i 6. �4 � 4i

7. �a � bi 8. �1 � 8i

9. �5 � i 10. �13i

Calculate each sum.

11. (�3 � i ) � (4 � 2 i ) 12. (8 � 12 i ) � (6 � 14i )

1 � 3i


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13. (9 � 3i ) � (7 � 3i ) 14. (�7 � 5i ) � (1 � 10i )

15. (8 � 2 i ) � (�4 � 2 i ) 16. (�7 � 5i ) � (7 � i )

Calculate each difference.

17. (�3 � i ) � (4 � 2 i ) 18. (8 � 12 i ) � (6 � 14i )

�7 � i

19. (9 � 3i ) � (7 � 3i ) 20. (�7 � 5i ) � (1 � 10i )

21. (8 � 2 i ) � (�4 � 2 i ) 22. (�7 � 5i ) � (7 � i )

Calculate each product.

23. (�3 � i )(4 � 2 i )

� �12 � 6i � 4i � 2 i 2 � �12 � 2 i � 2 � �14 � 2 i

24. (8 � 12 i )(6 �14i )

25. (9 � 3i )(7 � 3i )

26. (�7 � 5i )(1 � 10i )

27. (8 � 2 i )(�4 � 2 i )

28. (�7 � 5i )(7 � i )

29. (2 � 3i )(2 � 3i )


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30. (�6 � 4i )(�6 � 4i )

31. (10 � 10i )(10 � 10i )

32. (a � bi )(a � bi )

Calculate each quotient.

33. 9 � 2 i ______ 3 � 4i

9 � 2 i ______ 3 � 4i

� ( 9 � 2 i ______ 3 � 4i

) ( 3 � 4i ______ 3 � 4i

) � 27 � 36i � 6i � 8i 2 ___________________ 9 � 12 i � 12 i � 16i 2

� 27 � 30i � 8 ____________ 9 � 16

� 35 � 30i ________ 25

� 35 ___ 25

� 30i ___ 25

� 7 __ 5 � 6 __

5 i

34. 6 � 2 i ______ 8i

35. 3 � 6i ______ 4 � i

36. 8 � 5i _______ 10 � 2 i

Calculate the power of each complex number.

37. (�3 � 4i )3

� (�3 � 4i )(�3 � 4i )(�3 � 4i ) � (9 � 12 i � 12 i � 16i 2)(�3 � 4i )

� (9 � 24i � 16)(�3 � 4i ) � (�7 � 24i )(�3 � 4i ) � 21 � 28i � 72 i � 96i 2

� 21 � 44i � 96 � 117 � 44i


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38. (5 � i )3

39. (6 � 2i)3

40. (4 � 3i)3

Calculate the square root of each complex number.

41. √______

8 � 6i 42. √_______

5 � 12 i

√______

8 � 6i � a � bi

8 � 6i � (a2 � b2) � 2abi

8 � a2 � b2 and 6i � 2abi

a � 3 __ b

8 � ( 3 __ b

) 2 � b2

8 � 9 __ b2 � b2

8b2 � 9 � b4

b4 � 8b2 � 9 � 0

(b2 � 9)(b2 � 1) � 0

b2 � �9 or b2 � 1

b2 � 1

b � �1

a � 3 ___ �1

� �3

√______

8 � 6i � 3 � i or �3 � i


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43. √________

�3 � 4i 44. √________

�6 � 8i


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Complex Roots and ZerosComplex Roots and Simplifying Complex Roots

Vocabulary The types of solutions of a quadratic equation are shown. Match each solution to the form of its corresponding equation’s discriminant.

1. two distinct real solutions a. b2 � 4ac � 0

2. one real solution b. b2 � 4ac � 0

3. two distinct complex solutions c. b2 � 4ac � 0

Problem Set Calculate the coordinates of the vertex and the x-intercepts for each quadratic function.

1. f(x) � x2 � 4x � 2

a � 1, b � �4, c � 2

x � �b ____ 2a

� �(�4)

______ 2(1)

� 2

f(2) � 22 � 4(2) � 2

� 4 � 8 � 2

� �2

Vertex: (2, �2)

x-intercepts: (2 � 2 √__

2 , 0), (2 � 2 √__

2 , 0)

x � �b � √_________

b2 � 4ac ________________ 2a

� �(�4) � √

______________ (�4)2 � 4(1)(2) _______________________

2(1)

� 4 � √_______

16 � 8 ____________ 2

� 4 � √__

8 _______ 2 � 4 � 2 √

__ 2 ________

2

� 2 � √__

2 , 2 � √__

2


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2. f(x) � 0.5x2 � 7x � 12

3. f(x) � 2x2 � x � 6

4. f(x) � x2 � 2x � 8


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Use imaginary numbers to simplify each square root expression.

5. √_____

�27 6. √______

�400

� √_________

(�3)(3)(3)

� 3 √__

3 i

7. √______

�169 8. √_____

�54

Calculate the coordinates of the vertex and the complex zeros of each quadratic function. Check your solution.

9. f(x) � x2 � 4x � 13

a � 1, b � �4, c � 13 x � �b � √________

b2 – 4ac _______________ 2a

x � �b ____ 2 a

� �(�4)

______ 2(1)

� 2 � �(�4) � √

_______________ (�4)2 � 4(1)(13) ________________________

2(1)

f(2) � 22 � 4(2) � 13 � 4 � √________

16 � 52 _____________ 2

� 4 � 8 � 13 � 4 � √_____

�36 __________ 2 � 4 � 6i ______

2

� 9 x � 2 � 3i or x � 2 � 3i

Vertex: (2, 9) Zeros: x � 2 � 3i or x � 2 � 3i

Check:

f(2 � 3i) � (2 � 3i)2 � 4(2 � 3i) � 13

� (4 � 6i � 6i � 9i2) � 8 � 12i � 13

� 4 � 12i � 9 � 8 � 12i � 13

� 4 � 9 � 8 � 13 � 12i � 12i

� 0

f(2 � 3i) � (2 � 3i)2 � 4(2 � 3i) � 13

� (4 � 6i � 6i � 9i2) � 8 � 12i � 13

� 4 � 12i � 9 � 8 � 12i � 13

� 4 � 9 � 8 � 13 � 12i � 12i

� 0


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10. f(x) � �2x2 � 2x � 25


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11. f(x) � 2x2 � 6x � 9


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12. f(x) � x2 � 4x � 8


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Calculate the complex roots of each quadratic equation. Check your solution.

13. x2 � 2x � 5 � 0

a � 1, b � �2, c � 5 Check:

x � �b � √_________

b2 � 4ac ________________ 2a

(1 � 2 i)2 � 2(1 � 2 i) � 5

� �(�2) � √

______________ (�2)2 � 4(1)(5) _______________________

2(1) � (1 � 2 i � 2 i � 4 i2) � 2 � 4i � 5

� 2 � √_______

4 � 20 ____________ 2 � 1 � 4i � 4 � 2 � 4i � 5

� 2 � √_____

�16 __________ 2 � 2 � 4i ______

2 � 1 � 4 � 2 � 5 � 4i � 4i

� 1 � 2 i � 0

(1 � 2i)2 � 2(1 � 2i) � 5

� (1 � 2i � 2i � 4i2) � 2 � 4i � 5

� 1 � 4i � 4 � 2 � 4i � 5

� 1 � 4 � 2 � 5 � 4i � 4i

� 0

14. x2 � x � 1 � 0


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15. f(x) � �2x2 � 2x � 1


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16. f(x) � �x2 � 5x � 7


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Cases, Roots, and GraphsSolving Quadratic Inequalities

Vocabulary Describe how to solve a quadratic inequality using each of the given methods.

1. The Case Method

2. The Roots Method

3. The Graphing Method


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Problem Set For each quadratic function in factored form, show when f( x) � 0 and when f( x) � 0.

1. f(x) � (x � 1)(x � 2)

f( x) � 0

Case 1: x � 1 � 0 and x � 2 � 0

So, x � �1 and x � �2.

Case 2: x � 1 � 0 and x � 2 � 0

So, x � �1 and x � �2.

f( x) � 0

Case 1: x � 1 � 0 and x � 2 � 0

So, x � �1 and x � �2.

Case 2: x � 1 � 0 and x � 2 � 0

So, x � �1 and x � �2.

2. f(x) � (x � 3)(x � 4)

3. f(x) � (2x � 5)(x � 7)


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Name _____________________________________________ Date ____________________

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4. f(x) � (x � 4)(4x � 3)

Solve each quadratic inequality using the case method.

5. x2 � 3x � 0

x( x � 3) � 0

Case 1: Both factors are positive. Case 2: Both factors are negative.

x � 0 and x � 3 � 0 x � 0 and x � 3 � 0

x � 3 x � 3

x � 3 x � 0

Solution: x � 0 or x � 3

6. x2 � x � 12


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7. 3x2 � � x � 4

8. x2 � 5x � 2 � 16

Solve each quadratic inequality using the roots method.

9. x2 � 8x � 1 � 0

a � 1, b � �8, c � 1 Try: x � 0, 1, 10

x � �b � √_________

b2 � 4ac ________________ 2a

02 � 8(0) � 1 � 0 � 0 � 1 � 1 is NOT � 0

� �(�8) � √

______________ (�8)2 � 4(1)(1) ________________________

2(1) 12 � 8(1) � 1 � 1 � 8 � 1 � �6 � 0

� 8 � √___

60 ________ 2 � 4 � √

___ 15 102 � 8(10) � 1 � 100 � 80 � 1 � 21 is NOT � 0

x � 0.127, 7.873 Approximate Solution: 0.127 � x � 7.873

Exact Solution: 4 � √___

15 � x � 4 � √___

15


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10. �2 x2 � 8 � 6x

11. �3x2 � 10x � 0

12. x2 � 7x � 1 � 9


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Solve each quadratic inequality using the graphing method.

13. x2 � 4x � 12 y

86 102 4

2

−4

−2

−6

−8

−10

−12

−14

−16

−18

−2−4−6−8−10x

0 x-intercepts:

x2 � 4x � 12 � 0

( x � 2)( x � 6) � 0

x � 2 � 0 or x � 6 � 0

x � �2 x � 6

The portion of the parabola between the x-intercepts has y-values less than zero.

Solution: �2 � x � 6

14. � x2 � 3x � 4 � 0


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Name _____________________________________________ Date ____________________

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15. x2 � 8x � 0

16. 2 x2 � 5x � 3


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Name _____________________________________________ Date ____________________

Carl Friedrich Gauss: Child ProdigyArithmetic Sequences, Series, and Their Partial Sums

Vocabulary Explain how each set of terms is related by identifying their similarities and differences.

1. explicit formula and recursive formula

2. finite series and infinite series

3. arithmetic sequence and arithmetic series

Problem Set List the next three terms in each sequence and identify the common difference, d.

1. 7, 12, 17, 22, 27, 32, 37; d � 5

2. 1, �1, �3, �5

3. 6.4, 7.6, 8.8, 10

4. �7, �4, �1

5. �2, �8, �14, �20


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6. 8.5, 7, 5.5, 4

7. 3 __ 5 , 4 __

5 , 1, 6 __

5

8. 4, 17, 30, 43

Write an explicit formula for each sequence.

9. 9, 20, 31, 42, . . . 10. 16, 7, �2, �11, . . .

an � 11(n � 1) � 9

an � 11n � 11 � 9

an � 11n � 2

11. 33, 24, 15, 6, . . . 12. 13, 16, 19, 22, . . .

13. 6.4, 7.6, 8.8, 10, . . . 14. 100, 105, 110, 115, 120, . . .

Use the explicit formula to calculate a50.

15. an � 3n � 25 16. a

n � �2.5n � 14

3(50) � 25 � 150 � 25 � 175

17. an � �10n � 14 18. a

n � 1 __

2 n � 6

Write a recursive formula for each sequence.

19. 5, 7, 9, 11, . . . 20. 4, 7, 10, 13, . . .

an � an�1 � 2, a1 � 5

21. 12, 7, 2, �3, . . . 22. 4, 2, 0, �2, . . .


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23. 8.6, 7.3, 6, 4.7, . . . 24. 1 __ 3 , 1 __

6 , 0, � 1 __

6 , . . .

Use the explicit formula to calculate the first four terms of each arithmetic sequence.

25. an � 4n � 3 26. a

n � 1 __

3 (n � 1) � 6

7, 11, 15, 19

27. an � �2(n � 1) � 5 28. a

n � �6n � 20

29. an � 100(n � 1) � 50 30. a

n � �2n � 5

Use the recursive formula to calculate the first four terms of each arithmetic sequence.

31. an � a

n�1 � 9, a

1 � 6 32. a

n � a

n�1 � 9, a

1 � 6

6, �3, �12, �21

33. an � a

n�1 � 10, a

1 � 5 34. a

n � a

n�1 � 6, a

1 � 15

Write a recursive formula for each arithmetic sequence defined explicitly.

35. an � �5n � 20 36. a

n � n � 8

an � an�1 � 5, a1 � 15

37. an � 3(n � 1) � 2 38. a

n � 6(n � 1) � 3

39. an � �4(n � 1) � 1 40. a

n � �10(n � 1) � 25


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Write an explicit formula for each arithmetic sequence defined recursively.

41. an � a

n�1 � 6, a

1 � 20 42. a

n � a

n�1 � 19, a

1 � 1

an � 6(n � 1) � 20 � 6n � 14

43. an � a

n�1 � 2.4, a

1 � 4 44. a

n � a

n�1 � 3.5, a

1 � 7

Calculate each indicated sum for the given series.

4 5. 9 � 15 � 21 � 27 � 33 � 39 � 45

S4 � 72

S6 � 144

S � 189

46. 2 � 4 � 6 � 8 � 10 � 12 � 14 � 16 � 18 � 20

S4 �

S10

�

S �

47. (�3) � (�5) � (�7) � (�9) � (�11) � (�13)

S2 �

S4 �

S �

48. (�15) � (�10) � (�5) � 0 � 5 � 10 � 15

S2 �

S5 �

S �


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Name _____________________________________________ Date ____________________

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Use the explicit formula for the sequence to write a formula for each series using sigma notation.

49. 0 � 3 � 6 � 9 � 12 � 15

� i�1

6

(3i � 3)

50. (�7) � (�5) � (�3) � (�1) � 1

51. 19 � 14 � 9 � 4 � (�1) � (�6) � (�11) � (�16) � (�21) � (�26)

52. 100 � 90 � 80 � 70 � 60 � 50 � 40 � 30

Calculate the sum of the arithmetic series.

53. 9 � 17 � 25 � 33 � 41 � 49 � 57

S � 7(9 � 57)

_________ 2

� 7(66)

_____ 2 � 462 ____

2 � 231

54. 19 � 14 � 9 � 4 � (�1) � (�6) � (�11) � (�16) � (�21) � (�26)

55. (�36) � (�29) � (�22) � (�15) � (�8) � (�1) � 6 � 13 � 20 � 27 � 34

56. 0 � 1 __ 4 � 1 __

2 � 3 __

4 � 1 � 5 __

4 � 3 __

2 � 7 __

4 � 2 � 9 __

4 � 5 __

2 � 11 ___

4 � 3


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Name _____________________________________________ Date ____________________

Mathematics Empowers Us To Do Superhuman Mental Calculations!Modeling Partial Sums of Arithmetic Series with Quadratic Equations

Vocabulary Describe how to calculate the sum of the first n odd or even integers using each method.

1. Gauss’s Solution

2. a function

Problem Set Write an explicit formula for each arithmetic sequence.

1. 2, 8, 14, 20, 26, . . . 2. �5, �1, 3, 7, 11, . . .

an � 6(n � 1) � 2

an � 6n � 6 � 2

an � 6n � 4


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3. 43, 36, 29, 22, 15, . . . 4. 50, 45.5, 41, 36.5, 32, . . .

Write a function to represent the sum of the first n terms of each arithmetic sequence with the given explicit formula.

5. an � 6n � 4 6. a

n � 4n � 7

a1 � 2

Sn � n(a1 � an) _________

2

� n __ 2 (2 � 6n � 4)

� n __ 2 (6n � 2)

Sn � 3n2 � n

7. an � �n � 3 8. a

n � �8n � 8


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Calculate the sum of the first 20 terms of each arithmetic sequence by writing an explicit formula and using Gauss’s Solution method.

9. �11, �13, �15, �17, . . . 10. 9, 8, 7, 6, 5, . . .

an � �2n � 9

a1 � �11

a20 � �2(20) � 9 � �40 � 9 � �49

Sn � n(a1 � an) __________

2

S20 � 20(�11 � 49)

_____________ 2

S20 � 20(�60)

________ 2

S20 � �600

11. �5, 10, 25, 40, . . . 12. 12, 24, 36, 48, . . .


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Write a function to represent the sum of the first n terms of each arithmetic sequence with the given explicit formula. Use the function to calculate the sum of the first 100 terms of the sequence.

13. an � 3n � 5

a1 � �2

Sn � n(a1 � an) __________

2

� n __ 2 (�2 � 3n � 5)

� n __ 2 (3n � 7)

Sn � 3 __ 2 n2 � 7 __

2 n

Sn � 3 __ 2 n2 � 7 __

2 n

S100 � 3 __ 2 (100)2 � 7 __

2 (100)

S100 � 3 __ 2 (10,000) � 7 __

2 (100)

S100 � 15,000 � 350

S100 � 14,650

14. an � �2n � 26

15. an � 4n � 7


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Name _____________________________________________ Date ____________________

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16. an � �5n � 2

The first four figures in a block pattern of 100 figures are given. Use the figures to answer the questions.

17.

a. Write a sequence to represent the number of new blocks used in each figure.

1, 3, 5, 7, . . .

b. Write an explicit formula to calculate any term of the sequence.

an � 2(n � 1) � 1

an � 2n � 1

c. Find the 100th term in the sequence.

a100 � 2(100) � 1 � 199

d. Calculate the total number of new blocks used in the 10th figure.

a10 � 2(10) � 1 � 19

Sn � n(a1 � an) _________

2

S10 � 10(1 � 19)

__________ 2

S10 � 100

There are 100 new blocks in the 10th figure.


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18.

a. Write a sequence to represent the number of new blocks used in each figure.



d. Calculate the total number of new blocks used in the 10th figure.


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19.

a. Write a sequence to represent the number of new marbles used in each figure.



d. Calculate the total number of new marbles used in the 10th figure.


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20.

a. Write a sequence to represent the number of new marbles used in each figure.



d. Calculate the total number of new marbles used in the 10th figure.

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