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Álgebra II

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Respuestas R39
37. Circunferencia 39. Elipse 41. Hipérbola 43. Parábola 45. Elipse 47. Parábola 49. Circunferencia 51. Hipérbola 53. Parábola
55. Hipérbola 57. Parábola 59. Circunferencia 61. 63. 65. no, y otras 
respuestas también pueden funcionar. La razón debe ser 67. No, las gráficas de hipérbolas de esta forma no pasan la prueba de la recta
vertical 69. Dominio: (�q, �5] ´ [5, q); rango: 71. El eje transversal de ambas gráficas está a lo largo del eje x. Los vértices de la
segunda gráfica estarán más cercanos al origen y la segunda gráfica será más ancha. 73. Las respuestas variarán. 75.
76. 77. 78. 79. 80. 1
Conjunto de ejercicios 10.4 1. Un sistema no lineal de ecuaciones es un sistema en el cual al menos una ecuación no es lineal.
3. Sí, por ejemplo 5. Sí, por ejemplo 7. 9. 11.
13. 15. 17. No hay solución real 19.
21. 23. 25.
27. 29. 31.
33. 35. 37. No hay solución real. 39. No hay solución 
real. 41. Las respuestas variarán. 43. 20 metros por 22 metros 45. 9 pies por 30 pies 47. largo: 14 centímetros, ancho: 8 centímetros
49. 16 pulgadas por 30 pulgadas 51. L1.67 segundos 53. r � 6%, p � $125 55. L16 y L184 57. L5 y L95
59. (�1, �3), (3.12, �0.53) 61. 10 yardas, 24 yardas 63. Paréntesis, exponentes, multiplicaciones o divisiones, sumas o restas.
64. 65. 0.9 66. 67.
Ejercicios de repaso del capítulo 10 1. 2. 3. 4.
5. 6. 7. 8.
x�9
y
3
�7
(�3, �4)
x � �2(y � 4)2 � 3
x5
y
5
(4, 1)
�5
x � (y � 1)2 � 4
x3�7
y
5
(�3, �4)
�5
y � (x � 3)2 � 4
x7�3
y
7
�3
y � (x � 2)2 � 1
(2, 1)
18 L 2.83; 1-3, 4217; a -5, 
5
2
b5; a - 
5
2
, 3b13; a5
2
- 6b
k =
ln A - ln A0
t
51x + 2 + 15
x - 7
1x + 221x2
+ x + 12
115, 22, 115, -22, 1-15, 22, 1-15, -2213, 42, 13, -42, 1-3, 42, 1-3, -42
12, 12, 12, -12, 1-2, 12, 1-2, -1213, 02, 1-3, 02112, 122, 112, -122, 1-12, 122, 1-12, -122
1115, 121-115, 12, 1115, -12, 1-115, -1212, 02, 1-2, 0212, -42, 1-14, -202
10, -32, 115, 221-15, 2212, 2122, 12, -2122, 1-2, 2122, 1-2, -21221-1, 52, 11, 52
1-4, 112, a5
2
, 
5
4
b13, 02, a - 
9
5
, 
12
5
b13, -32, 1-3, 32
v = A2E
m
3x + 2
2x - 3
a -1, 
1
3
b-x2
- x + 11
y = - 
1
2
 x + 1
�
5
3
.
b
a
x2
18
-
y2
50
= 1
x2
9
-
y2
25
= 1,
x2
9
-
y2
36
= 1
y2
4
-
x2
16
= 1
31. a)
b)
x5�5
y
5
�5
9y2 � x2 � 9
y2
1
-
x2
9
= 1, y = ;
1
3
 x 33. a)
b)
x10�10
y
10
�10
25x2 � 9y2 � 225
x2
9
-
y2
25
= 1, y = ;
5
3
 x29. a)
b)
x5�5
y
5
�5
4y2 � 16x2 � 64
y2
16
-
x2
4
= 1, y = ;2x 35. a)
b)
x10�10
y 4y2 � 36x2 � 144
y2
36
-
x2
4
= 1, y = ;3x
9. a)
b)
x9
y
9
y � x2 � 8x � 22
(4, 6)
y = 1x - 422 + 6
10. a)
b)
x7�3
y
5
(6, �1)
�5
x � �y2 � 2y � 5
x = -1y + 122 + 6 11. a)
b)
x5�5
y
5
�5
x � y2 � 5y � 4
9 ,�
4
5
�
2
x = ay +
5
2
b 2
-
9
4 12. a)
b)
x10�10
y
15
(2, �32)
�35
y � 2x2 � 8x � 24
y = 21x - 222 - 32 13. a)
b)
x5�5
y
5
x2 � y2 � 42
�5
(0, 0)
x2
+ y2
= 42
14. a)
b)
x3�7
y
7
�3
(�3, 4)
(x � 3)2 � (y � 4)2 � 12
1x + 322 + 1y - 422 = 12 15. a)
b)
x5�5
y
7
�3
(0, 2)
x2 � y2 � 4y � 0
x2
+ 1y - 222 = 22 16. a)
b)
x7�3
y
3
�7
x2 � y2 � 2x � 6y � 1 � 0
(1, �3)
1x - 122 + 1y + 322 = 32 17. a)
b)
x9
y
9
x2 � 8x � y2 � 10y � 40 � 0
(4, 5)
1x - 422 + 1y - 522 = 12
53. No hay solución real 54. 55. 5 pies por 9 pies 56. y 57.
Examen de práctica del capítulo 10 1. Se forman al cortar un cono o un par de conos [10.1] 2. [10.1] 3. [10.1]a -1, 
3
2
b150 L 7.07
r = 3%, p = $4000L145L410, 132, 10, -132
R40 Respuestas
23. 24. 25. 26. 27. 28.
29. 30. 6p L 18.85 unidades cuadradas
x7�3
y
7
�3
(2, 1)
25(x � 2)2 � 9(y � 1)2 � 225
x3�7
y
5
�5
9
(x � 3)2
� 1�
25
y2
x7�3
y
3
�7 (3, �2)
4
(y � 2)2
16
(x � 3)2
� 1�
x5�5
y
5
9x2 � 16y2 � 144
�5
x2�2
y
1
4x2 � 9y2 � 36
�1
x10�10
y
10
�10
� 1�
6436
x2 y2
x5�5
y
5
�5
� 1�
94
x2 y2
18. a)
b)
x7�3
y
�9
x2 � y2 � 4x � 10y � 17 � 0
(2, �5)
1x - 222 + 1y + 522 = 111222 19. 20.
x4�4
y
5
�5
y � � 36 � x2
x5�5
y
7
�3
y � 9 � x2
21.
22. 1x - 522 + 1y + 322 = 9
1x + 122 + 1y - 122 = 4
31. a)
b)
x5�5
y
5
�5
� 1�
164
x2 y2
y = ;2x 32. a)
b)
x5�5
y
5
�5
� 1�
44
x2 y2
y = ;x 33. a)
b)
x10�10
y
10
�10
� 1�
364
y2 x2
y = ;
1
3
 x
34. a)
b)
x10�10
y
10
�10
� 1�
1625
y2 x2
y = ;
5
4
 x 35. a) b)
c)
x5�5
y
5
�5
x2 � 9y2 � 9
y = ;
1
3
 x
x2
9
-
y2
1
= 1 36. a) b)
c)
x10�10
y
10
�10
25x2 � 16y2 � 400
y = ;
5
4
 x
x2
16
-
y2
25
= 1
37. a) b)
c)
x10�10
y
8
4y2 � 25x2 � 100
y = ;
5
2
 x
y2
25
-
x2
4
= 1 38. a) b)
c)
x10�10
y
10
�10
49y2 � 9x2 � 441
y = ;
3
7
 x
y2
9
-
x2
49
= 1
39. Hipérbola 40. Elipse 41. Circunferencia
42. Hipérbola 43. Elipse 44. Parábola
45. Elipse 46. Parábola 47.
48. 49.
50. No hay solución real 51.
52. 14, 32, 14, -32, 1-4, 32, 1-4, -32
16, 02, 1-6, 02
1-3, 02, a - 
12
5
, 
9
5
ba5
2
, 
3
2
b
15, 02, 1-5, 02
4. [10.1]
x3�7
y
3(�3, 1)
�5
y � �2(x � 3)2 � 1
1-3, 12, 5. [10.1]
x4
y
5
(3, 1)
�5
x � y2 � 2y � 4 6. [10.1]
x3�7
y
3
�7
(�1, �2)
x � �y2 � 4y � 5
x � �(y � 2)2 � 1 7. [10.1]
x9
y
9
(2, 4)
(x � 2)2 � (y � 4)2 � 9
8. 9p L 28.27 unidades cuadradas [10.1] 9. [10.1] 10. [10.1] 11. [10.1]
x3�7
y
7
�3
x2 � y2 � 2x � 6y � 1 � 0
(�1, 3)
x5�5
y
5
�5
y � � 16 � x2
1x - 322 + 1y + 122 = 16
12. [10.2]
13. No, el eje mayor debe estar a lo largo del eje y. [10.2]
x2�2
y
5
4x2 � 25y2 � 100
�5
Respuestas R41
14. [10.2] 15. [10.2]
16. El eje transversal está a lo largo del eje correspondiente al término con coeficiente positivo en la ecuación en la 
forma estándar. [10.3] 17. [10.3]
18. [10.3] 19. [10.3] 20. Hipérbola, divida ambos lados de la ecuación entre 30. [10.3]
21. Elipse, divida ambos lados de la ecuación entre 100. [10.3]
22. [10.4]
23. No hay solución real [10.4]
24. 30 metros por 50 metros [10.4]
25. 5 pies por 12 pies [10.4]
Examen de repaso acumulativo 1. [1.5] 2. [2.1] 3. [2.1] 4. [2.6]
5. [3.1] 6. 139 [3.2] 7. (8, 6) [4.1] 8. [5.5] 9. base: 14 pies, altura: 8 pies [5.8]
10. [6.1] 11. [6.2] 12. 2 [6.4] 13. [7.2] 14. [7.5]
15. 3 [7.6] 16. [8.2] 17. 2 [9.6] 18. [9.7] 19. [10.2]
20. [10.3]
Capítulo 11
Conjunto de ejercicios 11.1 1. Una sucesión es una lista de números acomodados en un orden específico. 3. Una sucesión
finita es una función cuyo dominio incluye sólo a los primeros n números naturales. 5. En una sucesión decreciente, los términos 
disminuyen. 7. Una serie es la suma de los términos de una sucesión. 9. es la suma cuando i va de 1 a 5 de i � 4.
11. Es una sucesión creciente. Cada número en la sucesión es mayor que el número que le precede. 13. Sí, los signos de los términos alternan.
15. 6, 12, 18, 24, 30 17. 3, 7, 11, 15, 19 19. 21. 23. 25. 27. 31
29. 12 31. 1 33. 99 35. 37. 2, 15 39. 3, 17 41. 43. 45. 47. 64, 128, 256 49. 17, 19, 21 51.
53. 55. 57. 59. 17, 12, 7 61. 63.
65. 67. 69. 71. 73. 13 75. 169 77. 55 79. 25
81. 83. a) 6, 12, 18, 24 b) 85. Las respuestas variarán. 87. Las respuestas variarán. 89.
91. Sí, por ejemplo si n � 3, obtiene 93. a) 10 b) 11 c) 110 d) 29 e) No
94. 95. 96. 11 97.
Conjunto de ejercicios 11.2 1. En una sucesión aritmética, cada término difiere en una cantidad constante. 3. Se llama 
diferencia común. 5. Es un número positivo 7. Sí, por ejemplo 9. Sí, por ejemplo,
11. 4, 7, 10, 13, 16; 13. 15. 17. 100, 95, 90, 85, 80;
19. 14 21. 27 23. 12 25. 2 27. 9 29. 6 31. 33. 35. 37.
39. 4, 7, 10, 13; 41. 43.
45. 47. 100, 93, 86, 79; 49. 51. n = 11; s11 = 121n = 15, s15 = 330a10 = 37; s10 = 685
7
2
, 6, 
17
2
, 11; a10 = 26, s10 = 147.5
-8, -13, -18, -23; a10 = -53; s10 = -305-6, -4, -2, 0; a10 = 12; s10 = 30a10 = 31; s10 = 175
s11 = 407; d = 6s6 = 25. 5; d = 3.7s8 =
52
5
; d =
1
5
s10 = 100; d = 2
an = -5n + 105
1
2
, 2, 
7
2
, 5, 
13
2
; an =
3
2
 n - 17, 5, 3, 1, -1; an = -2n + 9an = 3n + 1
2, 4, 6, Á-1, -2, -3, Á
r = A V
ph
81y - 2x221y2
+ 2x2
 y + 4x422
5
4x1 + 4x2 + 4x3 = 41x1 + x2 + x32
©x = nxpn = 6nL42.83
a
3
i = 1
 
i2
4a
5
i = 1
 1i + 824 + 5 + 6 + 7 + 8 + 9 = 39
1
2
+ 2 +
9
2
+ 8 = 15
2 + 5 +10 + 17 + 26 + 37 = 972 + 5 + 8 + 11 + 14 = 40
1
16
, - 
1
32
, 
1
64
1
81
, 
1
243
, 
1
729
1, -1, 1
1
6
, 
1
7
, 
1
8
1
2
, 7-1, -10, 
13
20
81
25
4, -8, 16, -32, 64-1, 1, -1, 1, -1
3
2
, 
4
3
, 
5
4
, 
6
5
, 
7
6
7, 
7
2
, 
7
3
, 
7
4
, 
7
5
a
5
i = 1
 1i + 42
x10�10
y
10
�10
� 1�
1625
y2 x2
x1�1
y
2
9x2 � 4y2 � 36
�2
L2.31
2 ; i111
3
6x + 6y1x
x - y2
3y3>2
x1>2
2x2
- 9x - 5
21x + 321x - 42
x - 4
4x + 3
1x2
+ 621x2
- 72
x5�5
y
5
�5
y � �2x � 2
ex ` x 6 - 
5
3
 o x 7 1 f¤
19
4
-27x3
 y9
12, 132, 12, -132, 1-2, 132, 1-2, -132x5�5
y
5
�5
� 1�
94
x2 y2
x5�5
y � 1�
125
y2 x2
y = ;
7
4
 x
18, -72
x9
y
3
�7
(4, �2)
4(x � 4)2 � 36(y � 2)2 � 36
R42 Respuestas
53. 55. 57. 1275 59. 2500 61. 1395 63. 267 65. 42,372 67. 351
69. a) 27 b) 196 71. 73. 75. a) 19 pies b) 143.5 pies 77. 2 pies 79. a) 155 b) 780
81. $496 83. a) $45,600 b) $438,000 85. 93. 94.
95. 96.
Conjunto de ejercicios 11.3 1. Una sucesión geométrica es una sucesión en la que cada término después del primero es el
mismo múltiplo del término que le precede. 3. Para determinar la razón común, tome cualquier término, con excepción del primero, y 
divídalo entre el que le precede. 5. 0 7. Sí 9. Sí, s
q
, existe ya que 11. 2, 6, 18, 54, 162 13.
15. 17. 19. 21. 23.
25. 27. 128 29. 31. 128 33. 6144 35. 37. 39. 155 41. 7812 43. 10,160 45.
47. 49. 51. 53. 55. 57. 2
59. 61. 12 63. 65. 5 67. 6 69. 4 71. 24 73. 75. 77. 79. 81. 83.
85. 87. 89. a) 4 días b) L1.172 gramos 91. a) L330.78 millones b) L63.4 años
93. a) b) c) 95. 97. a) 28.512 pies b) 550 pies
99. a) 10.29 pulgadas b) 100 pulgadas 101. 211 103. a) $12,000, $9600, $7680 b) c)
105. 190 pies 107. a) y2 asciende más lentamente. b) 109.
110. 12 111.
112. 113.
114. 5 115. 9 metros, 12 metros
Examen de mitad de capítulo 1. [11.1] 2. 91 [11.1] 3. 1, 11 [11.1] 4. [11.1] 5. 45 [11.1]
6. [11.1] 7. [11.2] 8. [11.2] 9. 6 [11.2] 10. [11.2] 11. [11.2]
12. 11 [11.2] 13. 136 [11.2] 14. [11.3] 15. [11.3] 16. 315 [11.3] 17. [11.3] 18. 18 [11.3]
19. [11.3] 20. a) Una sucesión es una lista de números acomodados en un orden específico. b) Una sucesión aritmética es una sucesión
en la que cada término difiere por una cantidad constante. c) Una sucesión geométrica es una sucesión donde los términos difieren por un 
múltiplo común. d) Una serie es la suma de los términos de una sucesión. [11.1–11.3]
Conjunto de ejercicios 11.4 1. Las respuestas variarán. 3. 1 5. No, sólo se pueden determinar factoriales de números no 
negativos. 7. 14, el número de términos es uno más que el exponente. 9. 10 11. 1 13. 1 15. 70 17. 28 19.
21. 23. 25.
27. 29. 31.
33. 35. 37. Sí,
39. Sí, 41. o 43.
45. 47. (0, 10) 48. (10, 4) 49. 2, 9 50. 51.
Ejercicios de repaso del capítulo 11 1. 6, 7, 8, 9, 10 2. 3. 4. 5. 11 6. 4
1
5
, 
2
3
, 
9
7
, 2, 
25
9
6, 3, 2, 
3
2
, 
6
5
-1, 3, 9, 17, 27
f-11x2 =
x - 8
3
2x3
 y5130y1a + b2n = a
n
i = 0
 an
i
ban - i
 bi
x8, 24x7, 17,496x, 6561m = 0m = n17 - 32! = 17 - 3217 - 4217 - 52! = 4 # 3 # 2!
4! = 4 # 3!x16
- 24x14
 y + 252x12
 y2
- 1512x10
 y32187x7
- 5103x6
 y + 5103x5
 y2
- 2835x4
 y3
x10
+ 100x9
+ 4500x8
+ 120,000x71
16
 x4
-
3
2
 x3
+
27
2
 x2
- 54x + 8116x4
+ 16x3
+ 6x2
+ x +
1
16
243a5
- 405a4
 b + 270a3
 b2
- 90a2
 b3
+ 15ab4
- b5a4
- 4a3
 b + 6a2
 b2
- 4ab3
+ b48x3
- 36x2
+ 54x - 27
x3
+ 12x2
+ 48x + 64
29
33
- 
2
3
1
9
80, -40, 20, -10, 5
47 
1
2
3, -3-1-6, -1, 4, 9; an = -11 + 5na
5
i = 1
 a1
3
 i + 7b
-15, -19, -232, -1, -4, -7, -10
g-11x2 = 13 x - 9r =
S - 2a
S
6x3
- x2
 y - 16xy2
+ 6y3
n = 21; sn = 2,097,151
�10
�1
19
10
y2 y1
L$4915.20an = 12,000a4
5
bn - 1
L$15,938.48
1
128
L 0.78%an =
1
2
 a1
2
bn - 1
= a1
2
bn1
2
, 
1
4
, 
1
8
, 
1
16
, 
1
32
L$1.77r = 2 o r = -2; a1 = 7
r = 3; a1 = 5
17
33
8
9
8
33
-15-45
25
3
5
4
r =
2
3
; an =
3
4
 a2
3
bn - 1
r = -3; an = 21-32n - 1r = 2; an = 9122n - 1r =
1
2
; an = 3a1
2
bn - 1
- 
9279
625
- 
2565
256
50
729
1
64
- 
3
64
3, 
9
2
, 
27
4
, 
81
8
, 
243
16
1
3
, 
1
6
, 
1
12
, 
1
24
, 
1
48
5, -10, 20, -40, 80-1, -3, -9, -27, -8190, -30, 10, - 
10
3
, 
10
9
72, 24, 8, 
8
3
, 
8
9
6, -3, 
3
2
, - 
3
4
, 
3
8
ƒr ƒ 6 1, sq = 8
x�7
y
5
�5
(�4, 0)
(x � 4)2 � y2 � 25
312n - 5212n - 12
1-3, -52r =
A - P
Pt
an = 180°1n - 22
sn = n2101 # 50 = 5050
n = 29; s29 = 1421n = 17; s17 =
153
2
Respuestas R43
7. 8. 88 9. 10. 11. 12. 13.
14. 15. 16. 17.
18. 19. 20. 21. 29 22. 239 23. 132 24. 841
25. a) 10, 14, 18, 22 b) 26. a) 4, 10, 18, 28 b) 27. 5, 8, 11, 14, 17 28.
29. 30. 31. 30 32. 33. 34. 6 35.
36. 37. 38. 39.
40. 41. 42.
43. 44. 45. 46. 47. 6, 12, 24, 48, 96
48. 49. 50. 51. 52. 480 53. 216 54.
55. 441 56. 57. 58. 59. 60. 61.
62. 63. 10 64. 65. 66. 67. 32 68. 69. 70. 71. 72.
73. 74. 75.
76. 77. 15,050 78. 231 79. a) $36,000, $37,000, $38,000, $39,000
b) c) $41,000 d) $451,000 80. $102,400 81. a) b) c)
82. 83. 150 pies
Examen de práctica del capítulo 11 1. Una serie es la suma de los términos de una sucesión. [11.1] 2. a) Una sucesión
aritmética es aquella cuyos términos difieren en una cantidad constante. b) Una sucesión geométrica es aquella cuyos términos difieren en 
un múltiplo constante. [11.2–11.3] 3. [11.1] 4. [11.1] 5. [11.1]
6. 184 [11.1] 7. [11.1] 8. [11.3] 9. [11.2] 10. [11.3] 11. [11.2]
12. [11.2] 13. 12 [11.2] 14. [11.3] 15. [11.3] 16. [11.3] 17. 12 [11.3]
18. [11.3] 19. 56 [11.4] 20. [11.4] 21. 82 [11.2] 22. 91 [11.2]
23. $210,000 [11.2] 24. [11.3] 25. 364,500 [11.3]
Examen de repaso acumulativo 1. [2.2] 2. [3.5] 3. Un número infinito de soluciones [4.2]
4. [5.2] 5. [5.4] 6. [5.5] 7. [6.2]
8. 500 [6.6] 9. 3 [7.6] 10. 5 [7.6] 11. [8.1] 12. [8.2] 13. 5 [5.8]
14. [8.5] 15. [9.3] 16. [9.2] 17. [10.1]
18. [10.1] 19. [10.2] 20. 18 [11.3]
x5�5
y
5
9x2 � 16y2 � 144
�5
x3�5
y
3
�7
(�3, �1)
(x � 3)2 � (y � 1)2 � 16
1x + 622 + 1y - 222 = 49
x5�5
y
5
�5
y � 2x � 1
1
2
x10�10
y
2
�10
y � x2 � 4x
(2, �4)
3 ; 1309
30
-1 ; i114
5x2
+ 14x - 49
1x + 521x - 221a + b + 4221x2
+ 221x - 625x4
+ 29x3
+ 14x2
- 28x + 10
y = - 
11
3
 x +
38
3
b =
2A
h
L$851.66
x4
+ 8x3
 y + 24x2
 y2
+ 32xy3
+ 16y413
33
r =
1
3
; an = 15a1
3
bn - 139,063
5
256
243
-20
-40
5
12
, 
5
18
, 
5
27
, 
10
81
15, 9, 3, -3an = 5122n - 1an =
1
3
+
1
3
 1n - 12 =
1
3
 n
5 + 11 + 21 + 35 + 53 = 125s1 = 3; s3 =
181
36
- 
1
3
, 0, 
1
9
, 
1
6
, 
1
5
L$503.63
L$24,041.29L$2463.13L$2024.51an = $35,000 + 1000n
256a16
+ 3072a14
 b + 16,128a12
 b2
+ 48,384a10
 b3
x9
- 18x8
 y + 144x7
 y2
- 672x6
 y38x3
- 36x2
 y2
+ 54xy4
- 27y681x4
+ 108x3
 y + 54x2
 y2
+ 12xy3
+ y4
23
37
4
11
-12
25
6
27
2
-18-6
25
6
r =
2
3
; an =
9
5
 a 2
3
bn - 1
r =
1
3
; an = 10a1
3
bn - 1
r = 5; an = -4152n - 1r = 2; an = 6122n - 1127
8
585
8
- 
4305
64
4
243
2
27
-20, -4, - 
4
5
, - 
4
25
, - 
4
125
20, - 
40
3
, 
80
9
, - 
160
27
, 
320
81
-12, -6, -3, - 
3
2
, - 
3
4
n = 10; s10 = 180n = 11; s11 =
231
10
n = 7, s7 = 14n = 13, s13 = 442
-60, -55, -50, -45; a10 = -15, s10 = -375
5
6
, 
3
2
, 
13
6
, 
17
6
; a10 =
41
6
; s10 =
115
3
4, 1, -2, -5; a10 = -23, s10 = -95
-7, -3, 1, 5; a10 = 29, s10 = 110s9 = -42; d = - 
1
3
s7 =
48
5
; d =
2
5
s7 = -210; d = -6
s8 = 112; d = 2
1
2
-4-100, - 
499
5
, - 
498
5
, - 
497
5
, - 
496
5
1
2
, - 
3
2
, - 
7
2
, - 
11
2
, - 
15
2
5, 
14
3
, 
13
3
, 4, 
11
3
an = n1n + 32 = n2
+ 3npn = 4n + 6
1
2
+
2
3
+
3
4
+
4
5
=
163
60
1
6
+
4
6
+
9
6
+
16
6
+
25
6
=
55
6
6 + 14 + 24 + 36 = 80
10 + 13 + 18 = 41-3, -7, -11; an = 17 - 4n
16
7
, 
32
7
, 
64
7
; an =
2n - 1
7
- 
1
3
, 
1
9
, - 
1
27
; an = 1-12n134 - n2
32, 64, 128; an = 2ns1 = -9, s3 = -10s1 =
4
3
, s3 =
227
60
s1 = 9, s3 = 38s1 = 7, s3 = 27
26
81
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 /ENU <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>
 >>
>> setdistillerparams
<<
 /HWResolution [2400 2400]
 /PageSize [612.000 792.000]
>> setpagedevice
<<
 /ASCII85EncodePages false
 /AllowTransparency false
 /AutoPositionEPSFiles true
 /AutoRotatePages /None
 /Binding /Left
 /CalGrayProfile (Gray Gamma 1.8)
 /CalRGBProfile (Apple RGB)
 /CalCMYKProfile (U.S. Sheetfed Uncoated v2)
 /sRGBProfile (sRGB IEC61966-2.1)
 /CannotEmbedFontPolicy /Warning
 /CompatibilityLevel 1.4
 /CompressObjects /Off
 /CompressPages true
 /ConvertImagesToIndexed true
 /PassThroughJPEGImages true
 /CreateJDFFile false
 /CreateJobTicket false
 /DefaultRenderingIntent /Default
 /DetectBlends true
 /DetectCurves 0.1000
 /ColorConversionStrategy /LeaveColorUnchanged
 /DoThumbnails false
 /EmbedAllFonts true
 /EmbedOpenType false
 /ParseICCProfilesInComments true
 /EmbedJobOptions true
 /DSCReportingLevel 0
 /EmitDSCWarnings false
 /EndPage -1
 /ImageMemory 1048576/LockDistillerParams false
 /MaxSubsetPct 100
 /Optimize true
 /OPM 1
 /ParseDSCComments true
 /ParseDSCCommentsForDocInfo true
 /PreserveCopyPage true
 /PreserveDICMYKValues true
 /PreserveEPSInfo true
 /PreserveFlatness true
 /PreserveHalftoneInfo false
 /PreserveOPIComments false
 /PreserveOverprintSettings true
 /StartPage 1
 /SubsetFonts true
 /TransferFunctionInfo /Remove
 /UCRandBGInfo /Preserve
 /UsePrologue false
 /ColorSettingsFile (Color Management Off)
 /AlwaysEmbed [ true
 ]
 /NeverEmbed [ true
 ]
 /AntiAliasColorImages false
 /CropColorImages true
 /ColorImageMinResolution 150
 /ColorImageMinResolutionPolicy /OK
 /DownsampleColorImages true
 /ColorImageDownsampleType /Bicubic
 /ColorImageResolution 300
 /ColorImageDepth -1
 /ColorImageMinDownsampleDepth 1
 /ColorImageDownsampleThreshold 1.50000
 /EncodeColorImages true
 /ColorImageFilter /DCTEncode
 /AutoFilterColorImages true
 /ColorImageAutoFilterStrategy /JPEG
 /ColorACSImageDict <<
 /QFactor 0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /ColorImageDict <<
 /QFactor 0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /JPEG2000ColorACSImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /JPEG2000ColorImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /AntiAliasGrayImages false
 /CropGrayImages true
 /GrayImageMinResolution 150
 /GrayImageMinResolutionPolicy /OK
 /DownsampleGrayImages true
 /GrayImageDownsampleType /Bicubic
 /GrayImageResolution 300
 /GrayImageDepth -1
 /GrayImageMinDownsampleDepth 2
 /GrayImageDownsampleThreshold 1.50000
 /EncodeGrayImages true
 /GrayImageFilter /DCTEncode
 /AutoFilterGrayImages true
 /GrayImageAutoFilterStrategy /JPEG
 /GrayACSImageDict <<
 /QFactor 0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /GrayImageDict <<
 /QFactor 0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /JPEG2000GrayACSImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /JPEG2000GrayImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /AntiAliasMonoImages false
 /CropMonoImages true
 /MonoImageMinResolution 1200
 /MonoImageMinResolutionPolicy /OK
 /DownsampleMonoImages true
 /MonoImageDownsampleType /Bicubic
 /MonoImageResolution 1200
 /MonoImageDepth -1
 /MonoImageDownsampleThreshold 1.50000
 /EncodeMonoImages true
 /MonoImageFilter /FlateEncode
 /MonoImageDict <<
 /K -1
 >>
 /AllowPSXObjects false
 /CheckCompliance [
 /None
 ]
 /PDFX1aCheck false
 /PDFX3Check false
 /PDFXCompliantPDFOnly false
 /PDFXNoTrimBoxError true
 /PDFXTrimBoxToMediaBoxOffset [
 0.00000
 0.00000
 0.00000
 0.00000
 ]
 /PDFXSetBleedBoxToMediaBox true
 /PDFXBleedBoxToTrimBoxOffset [
 0.00000
 0.00000
 0.00000
 0.00000
 ]
 /PDFXOutputIntentProfile (None)
 /PDFXOutputConditionIdentifier ()
 /PDFXOutputCondition ()
 /PDFXRegistryName (http://www.color.org)
 /PDFXTrapped /Unknown
 /Description <<
 /JPN <FEFF3053306e8a2d5b9a306f30019ad889e350cf5ea6753b50cf3092542b308030d730ea30d730ec30b9537052377528306e00200050004400460020658766f830924f5c62103059308b3068304d306b4f7f75283057307e305930023053306e8a2d5b9a30674f5c62103057305f00200050004400460020658766f8306f0020004100630072006f0062006100740020304a30883073002000520065006100640065007200200035002e003000204ee5964d30678868793a3067304d307e305930023053306e8a2d5b9a306b306f30d530a930f330c8306e57cb30818fbc307f304c5fc59808306730593002>
 /FRA <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>
 /DEU <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>
 /PTB <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>
 /DAN <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>
 /NLD <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>/ESP <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>
 /SUO <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>
 /ITA <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>
 /NOR <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>
 /SVE <FEFF0041006e007600e4006e00640020006400650020006800e4007200200069006e0073007400e4006c006c006e0069006e006700610072006e00610020006e00e40072002000640075002000760069006c006c00200073006b0061007000610020005000440046002d0064006f006b0075006d0065006e00740020006d006500640020006800f6006700720065002000620069006c0064007500700070006c00f60073006e0069006e00670020006600f60072002000700072006500700072006500730073007500740073006b0072006900660074006500720020006100760020006800f600670020006b00760061006c0069007400650074002e0020005000440046002d0064006f006b0075006d0065006e00740065006e0020006b0061006e002000f600700070006e006100730020006d006500640020004100630072006f0062006100740020006f00630068002000520065006100640065007200200035002e003000200065006c006c00650072002000730065006e006100720065002e00200044006500730073006100200069006e0073007400e4006c006c006e0069006e0067006100720020006b007200e400760065007200200069006e006b006c00750064006500720069006e00670020006100760020007400650063006b0065006e0073006e006900740074002e>
 /ENU <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>
 >>
>> setdistillerparams
<<
 /HWResolution [2400 2400]
 /PageSize [612.000 792.000]
>> setpagedevice
<<
 /ASCII85EncodePages false
 /AllowTransparency false
 /AutoPositionEPSFiles true
 /AutoRotatePages /None
 /Binding /Left
 /CalGrayProfile (Gray Gamma 1.8)
 /CalRGBProfile (Apple RGB)
 /CalCMYKProfile (U.S. Sheetfed Uncoated v2)
 /sRGBProfile (sRGB IEC61966-2.1)
 /CannotEmbedFontPolicy /Warning
 /CompatibilityLevel 1.4
 /CompressObjects /Off
 /CompressPages true
 /ConvertImagesToIndexed true
 /PassThroughJPEGImages true
 /CreateJDFFile false
 /CreateJobTicket false
 /DefaultRenderingIntent /Default
 /DetectBlends true
 /DetectCurves 0.1000
 /ColorConversionStrategy /LeaveColorUnchanged
 /DoThumbnails false
 /EmbedAllFonts true
 /EmbedOpenType false
 /ParseICCProfilesInComments true
 /EmbedJobOptions true
 /DSCReportingLevel 0
 /EmitDSCWarnings false
 /EndPage -1
 /ImageMemory 1048576
 /LockDistillerParams false
 /MaxSubsetPct 100
 /Optimize true
 /OPM 1
 /ParseDSCComments true
 /ParseDSCCommentsForDocInfo true
 /PreserveCopyPage true
 /PreserveDICMYKValues true
 /PreserveEPSInfo true
 /PreserveFlatness true
 /PreserveHalftoneInfo false
 /PreserveOPIComments false
 /PreserveOverprintSettings true
 /StartPage 1
 /SubsetFonts true
 /TransferFunctionInfo /Remove
 /UCRandBGInfo /Preserve
 /UsePrologue false
 /ColorSettingsFile (Color Management Off)
 /AlwaysEmbed [ true
 ]
 /NeverEmbed [ true
 ]
 /AntiAliasColorImages false
 /CropColorImages true
 /ColorImageMinResolution 150
 /ColorImageMinResolutionPolicy /OK
 /DownsampleColorImages true
 /ColorImageDownsampleType /Bicubic
 /ColorImageResolution 300
 /ColorImageDepth -1
 /ColorImageMinDownsampleDepth 1
 /ColorImageDownsampleThreshold 1.50000
 /EncodeColorImages true
 /ColorImageFilter /DCTEncode
 /AutoFilterColorImages true
 /ColorImageAutoFilterStrategy /JPEG
 /ColorACSImageDict <<
 /QFactor 0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /ColorImageDict <<
 /QFactor0.15
 /HSamples [1 1 1 1] /VSamples [1 1 1 1]
 >>
 /JPEG2000ColorACSImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /JPEG2000ColorImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 30
 >>
 /AntiAliasGrayImages false
 /CropGrayImages true
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 >>
>> setdistillerparams
<<
 /HWResolution [2400 2400]
 /PageSize [612.000 792.000]
>> setpagedevice