ภาพถ่ายหน้าจอ 2565-05-17 เวลา 20.28.20

เอกสารประกอบการเรียนการสอน วิชาคณิตศาสตร์เพ่มิ พนู ค 22205 ช้ันมธั ยมศกึ ษาปีที่ 2 ลําดับและอนุกรม ช่อื -สกุล………………………………………………………ชน้ั ม.2/……..เลขท…ี่ …… ภาคเรยี นที่ 1 ปีการศึกษา 2565 กลุ่มสาระการเรยี นรู้คณิตศาสตร์ โรงเรยี นอัสสัมชัญศกึ ษา

(Sequence) 1, 2, 3, 4, ,n f(1), f(2), f(3), n f(1), f(2), f(3), , f(n), , f(n) 1, 2, 3, 4, f(1), f(2), f(3), , f(n) f(1), f(2), f(3), , f(n), f(1) a1 f(2) a2 f(3) a3 f(n) n an a1, a2, a3, ,an a1, a2, a3, ,an , 10, 20, 30, 40, a1 10,a2 20,a3 30,a4 40 an 10n a 1 = 1, a 2 = 3, a 3 = 5, a 4 = 7, a 5 = 9 1, 3, 5, 7, K a n = 2n - 1 1, 1 , 1 , 1 , 1 a1 = 1, a 2 = 1 , a3 = 1 , a4 = 1 , a5 = 1 2 3 4 5 2 3 4 5 1 an = n 2, 4, 8, 16, ... a 1 = 2, a 2 = 4, a 3 = 8, a 4 = 16 a n = 2n 1

a n = 5n - 2 a n = 5n - 2 a 1 = 5(1) - 2 = 3 a 2 = 5(2) - 2 = 8 a 3 = 5(3) - 2 = 13 a 4 = 5(4) - 2 = 18 3, 8, 13, 18 a n = 2n + 5 a n = 2n + 5 a 1 = 2(1) + 5 = 7 a 2 = 2(2) + 5 = 9 a 3 = 2(3) + 5 = 11 a 4 = 2(4) + 5 = 13 7, 9, 1 1, 1 3 a n = (- 2 )n a n = 4 + (- 1)n an = 2n an = (- 1)n 1+ n 2n - 1 a n = (- 2 )n a 1 = (- 2 )1 = - 2 a 2 = (- 2)2 = (- 2)( - 2) = 4 a 3 = (- 2)3 = (- 2) ( - 2)( - 2) = - 8 a 4 = (- 2 )4 = ( - 2) ( - 2) ( - 2) ( - 2) = 1 6 - 2, 4, - 8, 16 2

a n = 4 + (- 1)n a 1 = 4 + (- 1)1 = 4 + ( - 1 ) = 3 a 2 = 4 + (- 1)2 = 4 + 1 =5 a 3 = 4 + (- 1)3 = 4 + (- 1) = 3 a 4 = 4 + (- 1)4 = 4 + 1 =5 3, 5, 3, 5 an = 2n 1+ n a1 = 2(1) = 2 1+ 1 a2 = 2(2) = 4 1+ 2 3 a3 = 2(3) = 3 1+ 3 2 a4 = 2(4) = 8 1+ 4 5 2, 4 , 3 , 8 3 2 5 (- 1)n an = 2n - 1 a1 = (- 1 )1 = (- 1) = -1 2(1) - 1 1 a2 = (- 1)2 = 1 = 1 2(2) - 1 4- 1 3 a3 = (- 1)3 = (- 1) = - 1 2(3) - 1 6- 1 5 a4 = (- 1)4 = 1 = 1 2(4) - 1 8- 1 7 - 1, 1 , - 1 , 1 3 5 7 3

2, 6, 10, 14, ... 200, 195, 190, 185, ... 7 2 9, 2 4 3, 8 1, 2 7, ... 1, 4, 1 6, 6 4, ... 1 , 1 , 1 , 1 , ... 5 10 15 20 2 , 6 , 10 , 14 , 18 , 22 200 , 195 , 190 , 185 , 180 , 175 - - -- - 1 , 4 , 16 , 64 , 256 , 1024 xxx x x 729 , 243 , 81 , 27 , 9 , 3 1 , 1 , 1 , 1 , 1 , 1 5 10 15 20 25 30 + 1 + 1 + 1 + 1 + 1 5 5 5 5 5 11 25 30 4

an n { }n 1, 2, 3, ..., m 1, 2, 3, ..., m 5, 8, 1 1, 1 4, 1 7, 2 0 1 , 2 , 3 , 4 , 5 1, 3, 9, 2 7 2 3 4 5 6 5, 8, 1 1, 1 4, 1 7, 2 0 5 , 8 , 11 , 14 , 17 , 20 a 1 = 5 = 3(1) + 2 a 2 = 8 = 3(2) + 2 a 3 = 11 = 3(3) + 2 a 4 = 14 = 3(4) + 2 a 5 = 17 = 3(5) + 2 a 6 = 20 = 3(6) + 2 a n = 3n + 2 n 1,2,3,4, 5, 6 1 , 2 , 3 , 4 , 5 2 3 4 5 6 1 1 a1 = 2 = 1+ 1 a2 = 2 = 2 3 2+ 1 a3 = 3 = 3 4 3+ 1 a4 = 4 = 4 5 4+ 1 a5 = 5 = 5 6 5+ 1 n an = n+ 1 n 1,2,3,4, 5 5

1, 3, 9, 2 7 a 1 = 1 = 3 0 = 31- 1 a 2 = 3 = 31 = 32- 1 a 3 = 9 = 32 = 3 3- 1 a 4 = 27 = 33 = 34- 1 a n = 3n- 1 n 1,2,3,4 n an 1, 2, 3, ... an 1 , 2 , 3 , ..., n n 1, ... 2 3 4 + 1, 2, 3, 4, ... 7, 12, 17, 22, 27, ... 4, 8, 12, 16, 20, ... 1, - 1, - 3, - 5, - 7, ... 3, 0, - 3, - 6, - 9, ... 7 12 17 7, 12, 17, 22, 27, ... 22 27 6

7 12 17 22 27 5+ 2 10 + 2 15 + 2 20 + 2 25 + 2 5(2) + 2 5(3) + 2 5(4) + 2 5(5) + 2 5(1) + 2 a n = 5(n ) + 2 4, 8, 12, 16, 20, ... 4 8 12 16 20 4 8 12 16 20 4x1 4x2 4x3 4x4 4x5 an = 4n 1, - 1, - 3, - 5, - 7, ... 1 -1 -3 -5 -7 -- - - 7

1 -1 -3 -5 -7 (- 2) + 3 (- 4) + 3 (- 6) + 3 (- 8) + 3 (- 10) + 3 (- 2)(1) + 3 (- 2)(2)+ 3 (- 2)(3)+ 3 (- 2)(4)+ 3 (- 2)(5)+ 3 3 a n = (- 2)(n ) + 3 3, 0, - 3, - 6, - 9, ... 0 -3 -6 -9 --- - 3 0 -3 -6 -9 (- 3) + 6 (- 6) + 6 (- 9) + 6 (- 12) + 6 (- 15) + 6 (- 3)(1) + 6 (- 3)(2)+ 6 (- 3)(3)+ 6 (- 3)(4)+ 6 (- 3)(5)+ 6 a n = (- 3)(n ) + 6 8

an = an + b an = an2 + bn + c 1, - 1, - 3, - 5, - 7, ... 1 -1 -3 -5 -7 - -- - - an = an + b a1 = 1 = a + b ..........(1) a 2 = - 1 = 2a + b ..........(2) a = 1- b a = 1- b - 1 = 2(1- b )+ b 0 = 2 - 2b + b + 1 0 = 3- b b =3 a = 1- b a = 1- 3 = -2 a n = - 2n + 3 = 1 = -1 an = -3 a n = - 2n + 3 = -5 a 1 = - 2(1) + 3 = -7 a 2 = - 2(2) + 3 a 3 = - 2(3) + 3 a 4 = - 2(4) + 3 a 5 = - 2(5) + 3 a 1, a 2, a 3, a 4, a 5 a n = - 2n + 3 9

0, 5, 1 2, 2 1, 3 2, ... 32 0 5 12 21 an = an2 + bn + c n= 1 a1 = a + b + c =0 ..........(1) n= 2 ..........(2) n= 3 a2 = 4a + 2b + c = 5 ..........(3) ..........(4) a 3 = 9a + 3b + c = 12 ..........(5) a, b c an = n2 + 2n - 3 (2) - (1) 3a + b = 5 (3) - (2) 5a + b = 7 (5) - (4) 3a2a = 2 a= 1 a= 1 (4) 3(1) + b = 5 3(1) + b = 2 a = 1, b = 2 (1) 1+ 2+ c = 0 1+ 2+ c = - 3 an an = n2 + 2n - 3 a 1, a 2, a 3, a 4, a 5 a n = n2 + 2n - 3 a 1 = 12 + 2(1) - 3 = 0 a 2 = 22 + 2(2)- 3 = 5 a 3 = 32 + 2(3)- 3 = 12 a 4 = 42 + 2(4)- 3 = 21 a 5 = 52 + 2(5)- 3 = 32 10

an = 1 an = 2n + 3 a n = 5n- 1 n an = 1 an = n an = 1 - 1 n n- 2 5n- 1 3n 4n an = n2 + 1 an = (- 1)n+ 1 an = sin np 3n n(n + 1) 2 1, 8, 2 7, 6 4, 1 2 5 0.4, 0.04, 0.004, 0.0004 11

1, 1 , 1 , 1 - 1 , 1 , - 1 , 1 2 4 8 2 3 4 5 2, 2 1 , 3 1 , 4 1 ,5 1 lo g 1, lo g 1 0, lo g 1 0 0, lo g 1 0 0 0 2 3 4 5 1, 5, 9, 1 3, ... 5, 1 1, 1 7, 2 3, ... 12

2, 5, 10, 17, ... 2, 6, 12, 20, ... 2 , 4 , 6 , 8 , ... 3, 3, 3 3, 9, 9 3, ... 3 3 3 3 - 1, - 4, - 7, - 10, ... 2, 9, 22, 41, 66, ... 13

= 4, 11 7 = 4, 15 11 = 4, 19 15 = 4 n n+1 a1, a2, a3, a4, a5 an+1 an = d a1, a2, a3, a4, a5 n+1 n d 1, 8, 1 5, 2 2, ... d= 7 8 - 1 = 15 - 8 = 22 - 15 = 7 ,d= 4 4, 7, 1 1, 1 6, ... ,d= 2 7 4 11 7 - 12, - 8, - 4, 0, ... / - 8 - (- 12) = - 4 - (- 8) = 0 - (- 4) = 4 / 5, 7, 9, 1 1, ... / 7 - 5 = 9 - 7 = 11 - 9 = 2 / 7, 3, - 2, - 5, ... / 3 7 ( 2) 3 / / 2 2 14 , 8, 11, 14, 15, 3 4 4 -7, - 1 5 1, 3 , 2, 5 , 3, 7 ,... 2 2 2 2 -5 6 -2 0.5 7 8 -1, -3, -5, -7, - 9

a1 = 1 = 1+4 = 1 + 1(4) a2 = 5 = 1+4+4 = 1 + 2(4) a3 = 9 = 1+4+4+4 = 1 + 3(4) a4 = 13 = 1+4+4+4+4 = 1 + 4(4) a5 = 17 an = 1 + (n-1)4 = 1 + 4n - 4 = 4n - 3 a1, a2, a3, a4, a5 d a1 d a1 = a1 a2 = a1 + d a3 = a2 + d = (a1 + d) + d = a1 + 2d a4 = a3 + d = (a1 + 2d ) + d = a1 + 3d a5 = a4 + d = (a1 + 3d) + d = a1 + 4d an = an-1 + d = (a1 + (n-2)d) + d = a1 + (n-1)d n an = a1 + (n-1)d a1 = 2 , d = 3 , 5, 8, 11, 14 a6 = a5 + d = 14 + 3 = 17 a8 = a7 + d = 20 + 3 = 23 a7 = a6 + d = 17 + 3 = 20 a9 = a8 + d = 23 + 3 = 26 , 20, 23, 26 a1 = 1 , d = 7 an = a1 + (n-1)d an = 1 + (30-1)(7) an = 1 + (29)(7) an = 1 + 203 an = 204 15

, 3, -2, - f a1 = 8 , d = -5 a20 = a1 + 19d an = a1 + (n-1)d an = 8 + (n-1)(-5) an = 8 -5n + 5 an = -5n + 13 , a, b, c, d, e, f, g, 35 a1 = 3 , a9 = 35 an = a1 + (n-1)d a9 = a1 + 8d 35 = 3 + 8d 32 = 8d d =4 f an = a1 + (n-1)d a7 = a1 + 6d a7 = 3 + 6(4) a7 = 3 + 24 a7 = 27 f (2) - (1) a5 = 19 a20 = 64 an = a1 + (n-1)d a5 = a1 + 4d = a1 + 4d 64 = a1 + 19d 45 = 15d d=3 d=3 a1 + 4d = 19 a1 + 4(3) = 19 a1 = 19 - 12 a1 = 7 a1 = 7 16

a , a, 20 a - 6 = 20 - a a + a = 20 + 6 2a = 26 a = 13 , a, b, c, 44 a, b, c an +1 = an + d a=8+d b = a + d = (8 + d) + d = 8 + 2d c = b + d = (8 + 2d) + d = 8 + 3d 44 = c + d = (8 + 3d) + d = 8 + 4d 8 + 4d = 44 4d = 44 - 8 4d = 36 d=9 a = 8 + 9 = 17 b = 17 + 9 = 26 c = 26 + 9 = 35 an = a1 + (n-1)d a1 = 1001 , d = 7 , an = 1995 1995 = 1001 + (n-1)7 1995 - 1001 = (n-1)7 994 = (n-1)7 142 = n-1 n = 142 + 1 n = 143 17

2 . 4 10, 20, 40, 80, 5 -7, -5, -3, - 6 7 9 -10, -13, -16, -19, - ,1 2. a1 = - , -1, 3, 7, 1 ................... a1 . .. . . .. 18

, a, b, c, 29 a, b, c , a, b, c, d, e, f, g, h, i, 52 . f a6 = 15 a20 = 43 .. . 19

, a, b, c, 39 a, b, c ............ 20

6 2 , 12 2 , 24 2 , 48 2 n+1 n 3 6 12 24 n n+ 1 a1, a2, a3, a4, a5 an 1 r r a1, a2, a3, a4, a5 an 4 2, 8 2, 16 2 / 2 4 8 / / 2 9 3, 27 3, 81 3 3 9 27 / 4 10, 20, 30, 40, 5 -1, -4, -16, -64, - 20 4, 80 4, 320 4 5 20 80 20 2, 30 3 , 40 4 10 20 2 30 3 4 4, 16 4, 64 4 1 4 16 a1, a2, a3, a4, a5 n a1 r = (a1r)r = a1r2 a1 r = (a1r2)r = a1r3 a2 = a1r a3 = a2r a4 = a3r an = a1rn-1 a1 an = an 1 r n an a1rn-1 23

a1 = 5 , r = 4 a5 = a4r = 320(4) = 1280 a6 = a6r = 1280(4) = 5120 a7 = a6r = 5120(4) = 20480 , 5120, 20480 a1= 2 r= 3 2 a1 = 2 a2 = a1r = 2( 3 ) = 3 2 a3 = a2r = 3( 3 ) = 9 2 2 a4 = a3r = 9(3) = 27 4 22 2, 3, 9 , 27 24 an = a1rn-1 a1 = 2 , r = 3 a8 = a1r8-1 a8 = 2(37) a8 = 2(2187) a8 = 4374 a1 = 8 , r = 2 an = a1rn-1 an = 8(2n-1) an = 23(2n-1) an = 23+n-1 an = 2n+2 an = 2n+2 24

% ,000 ,000 + 100,000( 2 ) 100 100,000 + 100,000(0.02) 100,000(1 + 0.02) 100,000(1.02) 100,000(1.02) {100,000(1.02)( 2 )} 100 100,000(1.02) + 100,000(1.02)(0.02) 100,000(1.02)(1+0.02) 100,000(1.02)(1.02) 100,000(1.02)2 100,000, 100,000(1.02), 100,000(1.02)2 , a1 a2 a3 a11 an = a1rn-1 a11 = a1r10 a11 = 100,000(1.02)10 a11 100,000(1.218994) a11 121,899 ,899 25

1. , -4, 8, - 2 4 10, 20, 60, 120, 5 -1, 4, -16, 64, - 63 , 3,12, 48, ... 4 7 125, -25, 5, - 92 , 4 , 8 , 16 ,... 3 9 27 81 , 12, 16, 64 3 . a1 3 a1 3, r 2 a2 a1r a3 a2r (3)(2) 6 a4 a3r (6)(2) 12 (12)(2) 24 a1= 3 a1 3, r 2 r=4 . a1 5, r 3 . 27

a1 100, r 1 2 . 5. a1 a5 = a4 a6 = a6 a7 = a6 6. 1, -2, 4, - . . 10 a1 5, r 2 an a1r n 1 a10 a1r 9 a10 (5)(2)9 a10 (5)(512) a10 2560 a1 5, r 2 10 8. 7 a1 a7 = a1r7-1 an = a1rn-1 28

9. 8 . 10. 8 a1 5, r 3 9 . 12. 10 13. a1 12, r 3 . a1 12, r 3 . a1 4, r 1 2 . 29

14. 1 , 1 , 1 ,1,2,... 842 1 a1 1 r an 1 4 18 2 8 an 1 41 8 an a1r n 1 a1 1 ,r=2 8 . 15. 512 1,1,1, 64 32 16 . 3 729 28 . - . . 30

. 5 . 31 19. -4, 8, - . 20. 5 . . 21. 10 , 10 .

. 22. . 23. - - . . . 32

a 1, a 2, a 3 n a1 +a2 +a3 n 1, 3, 5, 7, 9 1+3+5+7+9 5, 10, 15, 20, 25, 30 5 + 10 + 15 + 20 + 25 + 30 2, 4, 8, 16, 32 2 + 4 + 8 + 16 + 32 3+8 5 -4 , -6, -8, -10 (-4) + (-6) + (-8) + (-10) ( 7, 14, 21, 28, 35, 42, 49, 56 5, 12, 22, 29, 36, 45 10, 18, 26, 34, 42, 50, 58, 66 -1, -4, -7, -10 a1, a2, a3, a4, a5, a6 n Sn n S n = a1 + a2 + a3 n S 5 = a1 + a2 + a3 + a4 +a5 S9 = a1 + a2 + a3 + a4 +a5 + a6 + a7 + a8 + a9 33

1 1, 3, 5, 7 ( Sn) 2 2, 9, 16, 23, 30, 37 3 5, 10, 15, 20, 25, 30, 35, 40 4 S4 = 1 + 3 + 5 + 7 = 16 6 S6 = 2 + 9 + 16 + 23 + 30 + 37 = 117 8 S8 = 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 = 180 ( Sn) ( 1 100, 90, 80, 70, 60, 50, 40 Sn n (a1 an ) 2 -2, -4, -6, -8, -10 2 3 -5, -3, -1, 1, 3, 5, 7 4 1, 8, 15, 22, 29, 36, 43, 50 5 75, 73, 71, 69 6 12, 15, 18, 21, 24, 27 n a1, a2 , a3 ,..., an Sn n Sn a1 a2 a3 ... an S359 12 + 33 =45 14 9 + 36 = 45 34 = 45 = 45

= 315 = 315 n Sn a1 a2 a3 ... an 2 an 1 an Sn a1 (a1 d ) (a1 2d ) ... {a1 (n 1)d} (1) Sn an an 1 an 2 ... a3 a2 a1 (2) + (3) : Sn an (an d ) (an 2d ) ... {an (n 1)d} n 2Sn (a1 an ) (a1 an ) (a1 an ) ... (a1 an ) n 2Sn n(a1 an ) Sn n (a1 an ) n 2 a1 = an = n 35

n Sn n (a1 an ) 2 n Sn n (a1 an ) 2 1 1+3+5+7 Sn n (a1 an ) n=4 2 a1 = 1 a4 = 7 S4 4 (a1 a4 ) 2 S4= 4 (1 7) 2 = (2)(8) = 16 1 + 3 + 5 + 7 = 16 2 4 + 7 + 10 + 13 + 16 + 19 + 22 Sn n (a1 an ) 2 n =7 S7 7 (a1 a7 ) 2 a1 = 4 S7 7 (4 22) a7 = 22 2 S7 7 (26) 2 S7 7(13) S7 91 4 + 7 + 10 + 13 + 16 + 19 + 22 = 91 5 + 10 + 15 + 20 + 25 + 30 + 35 Sn n (a1 an ) + 40 + 45 + 50 2 n = 10 S10 10 (a1 a10 ) a1 = 5 2 a10 50 S10 10 (5 50) 2 S10 5(55) S10 275 45 + 50 = 275 -- (-4) + 1 + 6 + 11 + Sn n (a1 an ) 16 + 21 2 n =8 a1 = -14 S8 8 (a1 a8 ) 2 a8 21 S8 4{( 14) 21} S8 4(7) S8 28 - (- - 36

n Sn n (a1 an ) 2 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 n n= a1 = Sn n (a1 an ) a9 = 2 (-2) +( -4) + ( -6) +( -8) + ( -10) (-10) + ( -7) + (-4) +(-1) + 2 + 5 + 8 + 11 3 + 10 + 17 + 24 + 31 + 38 + 45 +52 + 59 + 66 + 73 + 80 + 87 + 94 + 101 4 37

n Sn n {2a1 (n 1)d} 2 n sn n (a1 an ) n, a1, an 2 sn n [a1 {a1 (n 1)d}] an a1 (n 1)d 2 n, a1, d Sn n {2a1 (n 1)d} 2 1 20 sn n (a1 an ) 2 n = 20 , a1 = 7 n Sn 2 {2a1 (n 1)d} a20 n = 20 , a1 7 , d =8 a20 an a1 (n 1)d n 2 a20 7 (20 1)8 Sn {2a1 (n 1)d} a20 7 (19)(8) 20 2 a20 7 152 S 20 {2(7) (20 1)8} a20 159 S20 10{14 (19)8} Sn n (a1 an ) S 20 10(14 152) 2 S 20 20 (7 159) S 20 10(166) 2 S 20 1,660 S20 (10)(166) S20 1,660 20 1,660 38

2 sn n (a1 an ) n 2 a1 = 10 , an = 157 n n an a1 (n 1)d 157 10 (n 1)3 157 10 (n 1)3 147 (n 1)3 147 n1 3 49 n 1 49 1 n 50 n Sn n (a1 an ) Sn n {2a1 (n 1)d} 2 2 S 50 50 (10 157) S 50 50 {2(10) (50 1)3} 2 2 S50 25(167) S50 25{20 (49)3} S50 4,175 S50 25{20 147} S 20 25(167) S50 4,175 Sn ..................................................................................................................................................... 39

ให้นกั เรยี นนำความรู้เรือ่ งอนุกรมเลขคณติ คำนวณหาคา่ ต่อไปนี้ 30 40

12 600 a1 12, d 2, an 600 n n an a1 (n 1)d Sn n (a1 an ) 600 12 (n 1)2 2 600 12 (n 1)2 S 295 295 (12 600) 2 588 (n 1)2 S 295 295 (612) 2 588 n 1 2 S295 295(306) 294 n 1 S295 90,270 294 1 n 295 n 12 600 = 90,270 a1 105, d 7, an 994 , n an a1 (n 1)d = 70,336 994 105 (n 1)7 994 105 (n 1)7 889 (n 1)7 889 n 1 7 127 n 1 127 1 n 128 n Sn n (a1 an ) 2 S128 128 (105 994) 2 S128 64(1,099) S128 70,336 41

121 999 . 2. . 42

1n 1 3 Sn 3 Sn 3 Sn (2) - ( 1) 2 S n = 19,683 - 1 2 S n = 19,682 S n = 9,841 n a 1 + a1r + a1r2 + a1r 3 1r n 1 Sn = a 1 + a1r + a1r2 + a1r 3 1r n 1 r Sn = a1r + a1r2 + a1r 3 rSn - Sn = a1rn - a1 1r n 1 + a1rn (4) (3) r1 Sn ( r 1 ) = a1rn - a1 a1rn a1 Sn = r 1 a1 (rn 1) = r1 = a1( 1 rn ) (1 r) = a1 ( 1)(1 r n ) ( 1)(1 r) a1 (1 r n ) Sn = 1 r 43

n Sn = a1 (1 rn) 1 r a1r n = a1 r 1 a1rn 1r1 = a1 1r an = a1 r n - 1 r1 Sn = a1 anr 1r 3 1. Sn = a1 (rn 1) r1 2. Sn r1 r1 3. Sn rn) r1 = a1 (1 r 1 a1 anr = 1r 3 1 1+2 a1 = 1 , r = 2 , n = 8 Sn = a 1 rn a1 r 1 1(2)8 1 28 1 s8 = 21 = 1 256 1 255 1 255 2 1 , 1 , 1 , 1 ,......... .. 2 6 18 54 1 1 a1 = 2 , r 3 a1 (1 r n ) 1 (1 ( 1 )10 1 (1 ( 1 )10 ( 1 )( 3 ){1 ( 1 )10 } 1 r 2 1 3 2 3 2 2 3 Sn = = 1 = 2 3 1 33 4 3 Sn = {1 ( )10 } = 3 (1 591049) ( 3 )(5599004498) 14762 4 4 19683 44

3 1 1 .......... .......... ....... 1 1 4 8 256 1 1+ 2 a1 = 1 , r = 2 n an = a1r n-1 1 (1)( 1 ) n 1 256 2 1 1 ( 2 ) 8 ( 2 ) n 1 8 =n 1 , n=9 Sn = a1 (1 r n ) 1r 1[1 (1)9 ] 1 1 511 511 2 1022 255 2 512 512 512 1 512 256 S9 11 1 1 1 22 2 . . 1 .. 27 . . . .. .. .. 2 + (-6) +(18) + (-54) .. 45

2 + (-6) +(18) + (- .. . (-10) +(20) + (- .. . (-10) +(20) + (- .. . 1 1024 .. . . 4 4 31 4 32 ... 4 36 .. . 1 27 .. . 46