1

33102

2.1 n 1 n d

d = a 1n a n a n 1 = a n d n I

a 2 a1 = a 3 a 2 = a 4 a 3 = = a 1n a n

1. 1, 3, 5, ..., 99 ; d = 2 2. 6, 3, 0, ..., 27 ; d = 3

3. 2, 2 31 , 2 3

2 , 3, , 9 ; d = 31

a1 d

a1 , a1 d, a1 2d, a1 3d, , a1 (n1)d

2.1.1 1, 4, 7, 10, ... n 1 n d a 2 a1 = a 3 a 2 = a 4 a 3 = = a 1n a n = d

1, 4, 7, 10, ... a 2 a1 = 4 1 = 3 = d a 3 a 2 = 7 4 = 3 = d

2

a 4 a 3 = 10 7 = 3 = d (d) 3 d 3 2.1.2 a 2 a1 , a 3 a 2 , a 4 a 3

a 2 a1 a 3 a 2 a 4 a 3

1 3, 6, 12, 24, ... 2 12, 9, 6, 3, 0, 3 1, 1, 1, 1, 1, ..

a 2 a1 a 3 a 2 a 4 a 3

1 3, 6, 12, 24, ... 6 3 = 3 12 6 = 6 24 12 = 12

2 12, 9, 6, 3, 0, 9 12 = 3 6 9 = 3 3 6 = 3 3 1, 1, 1, 1, 1, .. 1 (1) = 2 (1) 1= 2 1(1) = 2

3

2.1

1 a 2 a1 , a 3 a 2 , a 4 a 3

a 2 a1 a 3 a 2 a 4 a 3

1 1, 9, 25, 49, ... 2 3, 6, 12, 18, ... 3 3, 4, 5, 6, ... 4 10, 5, 2

5 , 35 ,

5 7, 2, 3, 8, 6 21, 17, 13, 9, 7 x 3, x 6, x 9, x 12, 8 31, 27, 23, 18, 9

161 , 8

1 , 41 , 2

1 ,

10 0, 23 , 3, 2

9 , ...

4

2.2

a1 , 2a , a 3 , , a n , a 1n , a 2 a1 = a 3 a 2 = a 4 a 3 = = a 1n a n = d

2.2.1 1) 3, 5, 7, 9, ... 2) x, 2x, 5x,

1) 3, 5, 7, 9, ... d1 = 5 3 = 2

d 2 = 7 5 = 2

d 3 = 9 7 = 2

2 2) x, 2x, 5x, d1 = 2x x = 3x

d 2 = 5x (2x) = 3x 3x 2.2.2 p, 5p, 6p 9 P p, 5p, 6p 9 d = 5p p = (6p 9) 5p 4p = p 9 3p = 9 p = 3 p 3

5

2.2.3

1) 1, 2, 4, 7, 11, 16, ... 2) 6, 10, 14, 18, 22, ... 3) 5, 7, 9, 11, 13, ... 4) 13, 10, 7, 4, 1, ...

1) a 2 a1 = 2 1 = 1 a 3 a 2 = 4 2 = 2 a 2 a1 a 3 a 2

2) d = 4 3) d = 2 4) d = 3

6

2.2

2

1. 1.1 1, 4, 7, 10, ..., 3n 5, 3n 2, a 2 a1 = .. =

a 3 a 2 = .. = a 4 a 3 = .. = a n a 1n = .. =

1.2 1, 3, 5, 7, ..., 2n 2, 2n, a 2 a1 = .. =

a 3 a 2 = .. = a 4 a 3 = .. = a n a 1n = .. =

1.3 3, 10, 17, 24, ..., 7n 11, 7n 4, a 2 a1 = .. =

a 3 a 2 = .. = a 4 a 3 = .. = a n a 1n = .. =

1.4 5, 1, 3, 7, ..., 13 4n, 9 4n, a 2 a1 = .. =

a 3 a 2 = .. = a 4 a 3 = .. = a n a 1n = .. =

7

1.5 1, 23 , 2

4 , , n21 , )1n(2

1 , a 2 a1 = .. =

a 3 a 2 = .. = a 4 a 3 = .. = a n a 1n = .. =

2.

(d)

2.1 4, 10, 16, ..., 6n 2, 6 2.2 8, 6, 4, 2, 0, ... 2.3 24, 12, 6, 3, ... 2.4 5, 10, 15, 20, ... 2.5 1, 4, 9, 16, 25, ... 2.6 11, 2, 7, , 20 9n, 2.7 1, 1, 1, , (1) 1n , 2.8 3, 4, 5, , n + 2 2.9 2, 4, 6, , 200,

2.10 21 , 4

1 , 81 , , n2

1 ,

2.11 43 , 5

4 , 65 , , 3n

2n ,

2.12 21 , 2

3 , 25 , , 2

2n2 ,

2.13 a + 2, a + 4, a + 6, , a + 2n,

2.14 61 , 6

1 , 21 , , 3

n 21 ,

2.15 m, m + 5, m + 10, m + 15, , m + 75, 2.16 x 2y, x y, x, , x + 53y,

8

2.3 n n n n

n n n

a1 d

a1 , a1 + d, a1 + 2d, a1 + 3d, , a1 + (n 1)d

a1 = a1 + 0d = a1 + (1 1)d

a 2 = a1 + 1d = a1 + (2 1)d

a 3 = a1 + 2d = a1 + (3 1)d

a 4 = a1 + 3d = a1 + (4 1)d

: : a n = a1 + (n 1)d

a n = a1 + (n 1)d

na n 1a 1 d

2.3.1 6, 13, 20, 27, ... 1a = 6 d = 13 6 = 7 na = a1 + (n 1)d

na = 7)1n(6 = 7n76 = 1n7 na = 1n7

9

2.3.2 10, 4, 2, 8, ... n a1 = 10, d = 6 a n = a1 + (n 1)d a n = 10 + (n 1) (6) = 10 6n + 6 = 16 6n n 16 6n 1a d

10

2.3 n n

3

1 6, 3, 0, 3, 2 2, 2 3

1 , 2 32 , 3,

3 1, 4, 7, 10, 4 1, 10, 19, 28, 37, 5 5, 9, 13, 17, 6

43 , 4

23 , 443 ,

7 a, 2a+1, 3a+2, 8 x+5y, 2x+4y, 3x+3y, 9 10, 3, 4, 11, 10 4, 11, 18, 25,

11

2.4 n n () n n n n

1a d a n = a1 + (n 1)d

2.3 1a d a n = a1 + (n 1)d

na n 1a 1

d 2.4.1 4 21 51 355 na = d)1n(a1 4a = d)14(a1 = d3a1 51a = d)151(a1 = d50a1 d3a1 = 21 .(1) d50a1 = 355 .(2) (2) (1) d47 = 376

d = 47376

= 8

12

d = 8 (1)

3a1 ( 8 ) = 21 1a = 45 1a = 45 , d = 8 na = d)1n(a1 na = )8)(1n(45 = 45 + ( 8n) + 8 = n853 1a = 45 d = 8 na = n853

13

2.4 n n ()

3

1. 2. 3.

1. 3 5 6 16 a 3 = a 5 = .... a n = ................................................................................... ................................................................................................................ ................................................................................................................ ................................................................................................................ ................................................................................................................

................................................................................................................ 2. 2 9 4 17

a 2 = a 4 = a n a n = ................................................................................... ................................................................................................................ ................................................................................................................ ................................................................................................................ ................................................................................................................

14

................................................................................................................ 3. 6 20 25 81 n

a 6 = a 20 = . a n = ................................................................................... ................................................................................................................ ................................................................................................................ ................................................................................................................ ................................................................................................................

......................... ....................................................................................... ................................................................................................................

4. a 6 = 31 a10 = 51

a 6 = a10 = . a n = ................................................................................... ................................................................................................................ ................................................................................................................ ................................................................................................................ ................................................................................................................

......................... ....................................................................................... ................................................................................................................

15

2.5 n n n n

n + 1 n 1a d n na = d)1n(a1

n na = d)1n(a1 na n

1a 1 d

n n 2.5.1 5, 1, 3, 7, ... 18 a 1 = 5, d = 4, n = 18, a18

a n = a 1 + (n 1)d a18 = 5 + (18 1) (4) = 5 + 17(4) = 5 68 = 63 18 63

2.5.2 a1 10 d 4 a 25 a 1 = 10, d = 4, n = 25, a 25 a n = a1 + (n 1)d

a 25 = 10 + (25 1) (4) = 10 + 24(4) a 25 = 106 a 25 106

16

2.5.3 1 30 10 111 30 a1 = 30 , 10a = 111, 30a na = d)1n(a1 10a = d)110(a1 111 = d930 81 = 9d

d = 981

= 9

30a = 9)130(a1 30a = )929(30 = 26130 = 291 30 291

17

2.5 n n

4 n

........ 1. 7 21, 8, 5, ... a. 6 ........ 2. 20 5, 1, 3, 17,... b. 53 ........ 3. 44 10, 15, 20, ... c. 1,584 . 4. 50 16, 48, 80, d. 225

.. 5. 108 6, 6, 6, e. 529

.. 6. 1a = 6 , 20a = 101 100a f. 71

.. 7. 1a = 52 , d = 5

3 10a g. 57

. 8. 1a = 3 , d = 4 15a h. 501 .. 9. 1a = 4 , d = 6 10a i. 51 . 10. 1a = 9 , d = 2 22a j. 58

18

2.6 n n () n n n + 1 n 1a d n na = d)1n(a1

n na = d)1n(a1 na n

1a 1 d

n n

2.6.1 3 29 25 147 12 a 3 = 29 ; a 25 = 147 na = d)1n(a1 a1 + 2d = 29 . (1) a1 + 24d = 147 . (2) (2) (1) 22d = 176 d = 8 a1 = 45 a n = a1 + (n 1)d a12 = 45 + 11(8) = 45 88 = 43 12 43

19

2.6 n n ()

4 n

........... 1. 5a = 16 10a = 41 20a = 91

........... 2. 4a = 20 15a = 42 21a = 54

........... 3. 4a = 25 89a = 620 41a = 280

........... 4. 5a = 18 19a = 88 60a = 358

........... 5. 3a = 3x + 2 9a = 9x + 8 12a = 12x + 11

........... 6. 10a = 30 18a = 64 20a = 54

........... 7. 9a = 7 26a = 44 a16 = 14

........... 8. 12a = 37 50a = 151 42a = 120

........... 9. 1a = 2 4a = 13 20a = 93

........... 10. 3a = x + 6 8a = x + 16 12a = x 24

20

2.7 n + 1 n 1a d n na = d)1n(a1 n na = d)1n(a1 na n

1a 1 d

2.7.1 3 6, 13, 20, 27, 1a = 6 , d = 13 6 = 7 na = d)1n(a1 5a = 7)15(6 = 6 + (4 7) = 6 + 28 = 34 6a = 7)16(6 = 6 + (5 7) = 6 + 35 = 41 7a = 7)17(6 = 6 + (6 7) = 6 + 42 = 48 34, 41, 48

21

2.7

5

3

1. 7, 12, 17, 2. 17, 15, 13, 11, 3. 12, 9, 6, 3, 4. 20, 14, 8, 2, 5. 4, 9, 14, 19, 6. 5 , 2 5 , 3 5 ,

7. 91 , 9

20 , 941 ,

8. , 23 , 2 ,

9. a + 7, 2a + 5, 3a + 3, 10. 6a b, 5a, 4a + b,

22

2.8 na = d)1n(a1

n n = d)( + 1

2.8.1 7, 12, 17, 22, ... ,277 1 na = 1a + d)1n( 1a = 7, d = 5 , na = 227 277 = 7 + (n 1)5 277 = 7 + 5n 5 275 = 5n

5

275 = n

n = 55

2 n = d)( + 1

n = 15

7277

= 1

5270

= 54 + 1 n = 55

2 .8.2 10 1,000 13 1a = 13, d = 13 , na = 988

n = d)( + 1

n = 113

13988

n = 113975

n = 75 + 1 n = 76 10 1,000 13 76

23

2.8

6

1. 96, 83, 70, , 294 2. 24, 19, 14, 9, ... , 46 n 3. 4, 7, 10, 13, ... , 61 4. 2, 1, 4, 7, 10, ..., 64 5. 200 1,000 12 6. 40 80 7. 7 1,610 6 8. 4, 1, 6, ... , 181 9. 50 450 10. 100 200 8

24

2.9

a, A, b

A = 2ba

2.9.1 3 1) 3, x, 10 2) 5, x, 7 3) 11, x, 23 3

1) x = 2)103( = 2

13

2) x = 2)75( = 2

12 = 6

3) x = 2)2311( = 2

34 = 17

2.9.2 13, a, b, c, 1 a, b c

= 2113

= 7 = b

b = 7 13, a, 7, c, 1

a = 2713 = 10

13, 10, 7, c, 1

c = 217 = 4

a = 10 , b = 7 , c = 4

25

2.9

7

......... 1. 31, x, 25 a. 7 .. 2. x, ___, 3x+2 b. x + 3 . 3. x, ___, x+6 c. 15 .. 4. x y z 1, x, y, z, 25 d. 57 . 5. 5, a, b, c, 13 a e. 1,221 .. 6. 7, a, b, c, 31 a+b+c f. 3

.. 7. 5, a, b, 7 ab g. 15

.. 8. 9, x, y, z, 7 yxz h. 28

.. 9. 5, 10, x x i. 5 . 10. 8, x, y, 13 x + y j. 2x+1

26

2.10 n na = d)1n(a1 na n

1a 1 d 2.10.1 10,000 500 10 10,000, 10,500 , 11,000, 1a = 10,000 , d = 500, n = 11 na na = d)1n(a1 11a = 10,000 + (111)500 = 10,000 + (10)500 = 15,000 10 15,000

27

2.10.2 1 2 3 3 5 29 1. 2. 29 3

1. 1, 3, 5, , 29 1a = 1 , na = 29 d = 2 n na = d)1n(a1 29 = 1 + )1n( 2 29 = 1+ 2n 2 30 = 2n n = 15 15

2. 1) n 1 3 n 2 4 n 3 5 : : n 15 17 29 17

28

2.10.3 1,000,000 20% 20% 1 1,000,000

2 10080000,000,1

3 10080

10080000,000,1

1,000,000 , 10080000,000,1 , 100

8010080000,000,1 ,

a1 = 1,000,000 10080r

an = ra 1n1

a6 = 16

10080000,000,1

= 510080000,000,1

a6 = 327,860 327,860

29

2.10

8

1. 2. 3.

1. 3 100 500

................................................................................................. a n = ............................................................................................................. ................................................................................................................ ................................................................................................................ ................................................................................................................ ................................................................................................................ ................................................................................................................

2. 20, 16 12 96 ................................................................................................. a n = ............................................................................................................ ................................................................................................................ ................................................................................................................ ................................................................................................................ ................................................................................................................ ................................................................................................................

30

3. 3 70 376 ............................................................................................................. a n = ........................................................................................................................... .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. ..............................................................................................................................

4. 3 61 379 ................................................................................................. a n = ......................................................................................................................... ............................................................................................................................ ............................................................................................................................ .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. ............................................................................................................................

31

5. 24,000 2,000 2,000 10

................................................................................................. a n = ....................................................................................................................... .......................................................................................................................... .......................................................................................................................... .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. ..........................................................................................................................

6. 1 50,000 10,000 8

................................................................................................. a n = ..................................................................................................................... ........................................................................................................................ .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................. .............................................................................................................................