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  • Topic: Geometric progression, Sequence, Geometric series
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TOPIC 4
GEOMETRIC SEQUENCE
Geometric sequence or geometric progression-is a sequence in which a term is obtained by multiplying the preceding term by a constant number, called the common ratio, r. * A sequence in which the terms differ by a constant ratio.

Like the arithmetic sequence, each of the terms in a geometric sequence is related to the preceding term through a definite pattern. SUBTOPIC 1: DEFINITION of GEOMETRIC SEQUENCE
Study the given sequences and see if you can get the pattern or rule. A. 3,6,12,24,48,C.6,18,54,162
B. 200,100,50,55D.3,-6,12,-24,48
SUBTOPIC 2: DESCRIBING GEOMETRIC SEQUENCE AND FINDING THE COMMON RATIO If we denote the first term a1 and the common ratio as r, then the terms of a geometric sequence are: a1,a1r1,a1r2,a1r3, ..., a1rn-1

Take note how the common ratio is obtained. Using example A: second termfirst term=42=2sixth termfifth term=6432=2
fourth termthird term=168=2
That is, the common ratio is obtained by dividing any term by the preceding term. Illustrative examples:
Give the next four terms in each sequence:
A. -3,-6,-12. . .B. 2,8,32
a1=-3; r = 2 a1= 2 ; r= 4
an= a1 rn-1 an= a1 rn-1
a4= -3(2)4-1 a4= 2(4)4-1
a4= -3(2)3 a4= 2(4)3
a4= -3(8) a4= 2(64)
a4 = -24 a4= 128
a5= -3(2)5= -48 a5= 2(4)5= 512
a6= -3(2)6= -96 a6= 2(4)6= 2048
a7= -3(2)7= -192 a7= 2(4)7= 8192
SUBTOPIC 3: FINDING THE nth TERM OF A GEOMETRIC SEQUENCE
Given the first term and the common ratio of a geometric sequence, the nth term can be found using the formula an=a1rn-1. The nth term of a geometric sequence is an=a1rn-1, Where a1= the first term

an= the nth term
r= the common ratio.

Illustrative examples:
A. Find the 8th term of the geometric sequence 80,240,720. . . an= a1rn-1
an= 8; a1= 80; r=3
a8= 80(3)8-1
a8= 80(3)7
a8= 80(2187)
a8= 174, 960

B. Find the r of the geometric sequence if the 1st term is 34 and the 4th term is 814. . .

an= a1rn-1
an= 814; a1= 34; n= 4; r=?
a4= 34(r)4-1
814= 34(r)3
r3= 814 x 43= 27 r=327 r= 3

TOPIC 5
Geometric series- a series whose associated sequence is geometric. GEOMETRIC SERIES

A geometric series can be found in much the same way as the arithmetic series. A corresponding formula can be worked out. SUBTOPIC 1: THE FORMULA FOR A FINITE GEOMETRIC SERIES
Study how the formula for a geometric series is obtained.
A geometric series Sn can be written as Sn= a1 + a1r + a1r2 +. . . + a1rn-1. (1) Multiply both sides of the equality by r. rSn= a1r + a1r2 + a1r3 +. . .+ a1rn-1 + a1rn. (2) Subtract equation (2) from equation (1). Factoring and solving for Sn gives: Sn-rSn= a1-a1rn

Sn (1-r)= a1-a1rn
Sn= a-a r1-r or Sn= a (1-r )1-r
The formula for the sum of the first n terms in a geometric sequence is Sn=a-a r1-r or Sn= a (1-r )1-r
Where Sn= the sum
a1= the first term
an= the common ratio, r ≠ 1.

Where r ≠ 1

Illustrative example:
A. N= 6; a1= 12; r= -3S6= 12-87484
Sn= a-a r1-rS6= 87364
S6= 12-12(-3)1-(-3) S6= 2179 S6= 12-12(729)4 S6= 12-12(729)4
B. N= 4; a1= 20; r= -4
Sn= a-a r1-r
S4=20-20(-4)1-(-4)
S4= 20-20(264)5
S4= 20-20(264)5
S4= 20-52805
S4= 52605
S4= 1052

SUBTOPIC 2: INFINITE GEOMETRIC SERIES
Infinite geometric series is the indicated sum of the terms of an infinite geometric sequence. The series 1+2+4+8+16+... is an example of infinite...
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