Barisan dan DeretGeometri

By: Jeffrey Bivin

Lake Zurich High School

jeff.bivin@lz95.orgB

Last Updated: October 11, 2005

Barisan Geometri

1, 2, 4, 8, 16, 32, … 2n-1, …

3, 9, 27, 81, 243, … 3n, . . .

81, 54, 36, 24, 16, … , . . .1

3

281

n

5

1

1

14

3

2

3

23

n

n

n

n

Jeff Bivin -- LZHS

Barisan Geometri Suku Ke-n

Un = a·r(n-1)

Jeff Bivin -- LZHS

Menentukan Rumus Barisan GeometriSuku Pertama 2

Rasio 3

un = a·r(n-1)

un = 2(3)(n-1)

Jeff Bivin -- LZHS

Tentukan rumus barisan geometri Un

Suku Pertama 128

Rasio (1/2)

1

2

1128

n

nU

17

2

12

nnU

Un = a·r(n-1)

82

1nnU

Jeff Bivin -- LZHS

Tentukan rumus barisan geometri suku ke- n atau Un

Suku pertama 64

Rasio (3/2)

1

2

364

n

nU

1

16

2

32

n

n

nU

Un = a·r(n-1)

7

1

2

3

n

n

nU

Jeff Bivin -- LZHS

Menentukan Nilai Suku Ke-103, 6, 12, 24, 48, . . .

a = 3

r = 2

n = 10Un = a·r(n-1)

U10 = 3·(2)10-1

U10 = 3·(2)9

U10 = 3·(512)U10 = 1536

Jeff Bivin -- LZHS

Menentukan Suku Ke-82, -10, 50, -250, 1250, . . .

a= 2

r = -5

n = 8Un = a·r(n-1)

Un = 2·(-5)8-1

Un = 2·(-5)7

Un = 2·(-78125)

Un = -156250

Jeff Bivin -- LZHS

Menjumlahkan Deret Geometri

r

raaraS

nn

i

nn

1

.

1

1

Jeff Bivin -- LZHS

1 + 3 + 9 + 27 + 81 + 243

a = 1

r = 3

n = 6 31

311 6

nS

2

7291

nS

3642

728

nS

r

raaS

n

n

1

Jeff Bivin -- LZHS

4 - 8 + 16 - 32 + 64 – 128 + 256

a = 4

r = -2

n = 7 )2(1

)2(44 7

nS

3

)128(44 nS

1723

516

3

5124

nS

r

araS

n

n

1

Jeff Bivin -- LZHS

Alternative Sum Formula

r

araS

n

n

1

1 nn arU

rarrU nn 1

nn arrU

r

rUaS nn

1

Kita tahu bahwa:

Dikalikan r:

Sederhanakan:

Substitusikan:

Jeff Bivin -- LZHS

Tentukan jumlah barisan geometri

32

1

32

729256

5

nS

312187512

5 nS

729

10423nS

r

rUaS nn

1

729

256nU

3

2r

5a

Jeff Bivin -- LZHS

Hitunglah

a = 2

r = 2

n = 10

Un = 102421

222 10

nS

1

102422

nS

20461

2046

nS

r

raaS

n

n

1

10

1

2k

k= 2 + 4 + 8+…+1024

Jeff Bivin -- LZHS

Hitunglah

a = 3

r = 2

n = 8

Un = 38421

233 8

nS

1

25633

nS

7651

765

nS

r

araS

n

n

1

8

1

123j

j= 3 + 6 + 12 +…+ 384

Jeff Bivin -- LZHS

Rangkuman -- Geometri

r

rUaS nn

1

Rumus Suku-n

Jumlah Suku-n

r

araS

n

n

1Un = a·r(n-1)

Jeff Bivin -- LZHS

Deret Geometri Tak Hingga

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The Magic Flea(magnified for easier viewing)

There is no flea like a Magic Flea

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The Magic Flea(magnified for easier viewing)

1...32

1

16

1

8

1

4

1

2

1S

21

41

81

161

321...

Jeff Bivin -- LZHS

Jumlah Deret Geometri Tak Hingga

r

araS

i

n

11

1

1runtuk

Jeff Bivin -- LZHS

21

21

1S

121

21

S

r

aS

1

21a

21r

1...32

1

16

1

8

1

4

1

2

1S

Ingat --The Magic Flea

Jeff Bivin -- LZHS

311

6

S

96

32

S

r

aS

1

...27

2

9

2

3

226 S

6a

3

1r

Jeff Bivin -- LZHS

A Bouncing Ballrebounds ½ of the distance from which it fell --

What is the total vertical distance that the ball traveled before coming to rest if it fell

from the top of a 128 feet tall building?

Jeff Bivin -- LZHS

A Bouncing Ball

Lintasan Turun = 128 + 64 + 32 + 16 + 8 + …

256128

1

128

1 21

21

r

aS

Jeff Bivin -- LZHS

A Bouncing Ball

Lintasan Naik = 64 + 32 + 16 + 8 + …

12864

1

64

1 21

21

r

aS

Jeff Bivin -- LZHS

A Bouncing Ball

L Naik = 64 + 32 + 16 + 8 + … = 128

L Turun = 128 + 64 + 32 + 16 + 8 + … = 256

TOTAL = 384 ft.

Jeff Bivin -- LZHS

A Bouncing Ballrebounds 3/5 of the distance from which it fell --

What is the total vertical distance that the ball traveled before coming to rest if it fell from the top

of a 625 feet tall building?

Jeff Bivin -- LZHS

A Bouncing BallL Turun = 625 + 375 + 225 + 135 + 81 + …

5.1562625

1

625

1 52

53

r

aS

Jeff Bivin -- LZHS

A Bouncing BallL Naik = 375 + 225 + 135 + 81 + …

5.937375

1

375

1 52

53

r

aS

Jeff Bivin -- LZHS

A Bouncing Ball

L Naik = 375 + 225 + 135 + 81 + … = 937.5

L Turun = 625 + 375 + 225 + 135 + 81 + … = 1562.5

TOTAL = 2500 ft.

Jeff Bivin -- LZHS

1.1

9.

S

19.

9.S

r

aS

1

...00009.0009.009.09.9. nS

9.a

1.r

Menentukan jumlah tak hingga

Jeff Bivin -- LZHS

Pecahan – Desimal Tak hingga

911.

922.

31

933.

944.

955.

32

966.

977.

988.

19. 99

Jeff Bivin -- LZHS

Mari kita coba kembali

3

13.

3

13.

3

13.

13

39.

Jeff Bivin -- LZHS

Keunikan

9.xlet9.910 x

subtract 99 x19

9 x

Jeff Bivin -- LZHS

1.1

9.

S

19.

9.9. S

r

aS

1

1

...00009.0009.009.09.9.9.

9.1 a

1.r

OK now a series

Jeff Bivin -- LZHS

That’s All Folks

19.

.9 = 1.9 = 1

Jeff Bivin -- LZHS