Download - Barisan dan Deret Geometri
Barisan dan DeretGeometri
By: Jeffrey Bivin
Lake Zurich High School
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
Jeff Bivin -- LZHS
The Magic Flea(magnified for easier viewing)
There is no flea like a Magic Flea
Jeff Bivin -- LZHS
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