TOPIC four

GEOMETRIC SEQUENCE

Geometric sequence or geometric progression-is a sequence in which a term is obtained by multiplying the preceding term by a continuous number, referred to as the common ratio, r. * A sequence where the terms differ by a frequent ratio.

Like the arithmetic sequence, each of the terms in a geometric sequence relates to the previous term by using a definite design. SUBTOPIC 1: DEFINITION of GEOMETRIC SEQUENCE

Research the offered sequences and discover if you can get the pattern or perhaps rule. A. 3, six, 12, twenty four, 48, C. 6, 18, 54, 162

B. 2 hundred, 100, 60, 55D. several, -6, doze, -24, forty eight

SUBTOPIC 2: DESCRIBING GEOMETRIC SEQUENCE AND FINDING THE COMMON RATIO If we denote the first term a1 plus the common proportion as r, then the terms of a geometric sequence happen to be: a1, a1r1, a1r2, a1r3,..., a1rn-1

Keep in mind how the prevalent ratio is usually obtained. Using example A: second termfirst term=42=2sixth termfifth term=6432=2

fourth termthird term=168=2

That is, the most popular ratio can be obtained by dividing any kind of term by preceding term. Illustrative cases:

Provide the next 4 terms in each pattern:

A. -3, -6, -12... N. 2, almost 8, 32

a1=-3; r sama dengan 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 COLLECTION

Given the first term and the prevalent ratio of your geometric collection, the nth term are available using the solution 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 most popular ratio.

Illustrative examples:

A. Find the 8th term of the geometric sequence 70, 240, 720... an= a1rn-1

an= almost eight; a1= 70; r=3

a8= 80(3)8-1

a8= 80(3)7

a8= 80(2187)

a8= 174, 960

B. Get the l of the geometric sequence in case the 1st term is 34 and the next 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 five

Geometric series- a series whose associated pattern is geometric. GEOMETRIC SERIES

A geometric series are available in much the same approach as the arithmetic series. A corresponding formula may be worked out. SUBTOPIC 1: THE FORMULA TO GET A FINITE GEOMETRIC SERIES

Examine how the method for a geometric series is definitely obtained.

A geometrical series Sn can be created as Sn= a1 + a1r & a1r2 +... + a1rn-1. (1) Multiply both sides in the equality by simply r. rSn= a1r + a1r2 + a1r3 &... + a1rn-1 + a1rn. (2) Take away equation (2) from formula (1). Financing 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 solution for the sum of the first and terms within a geometric collection is Sn=a-a r1-r or perhaps Sn= a (1-r )1-r

Where Sn= the total

a1= the initially term

an= the most popular ratio, ur в‰ 1 )

Where r в‰ you

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: ENDLESS GEOMETRIC SERIES

Infinite geometric series is a indicated total of the conditions of an endless geometric sequence. The series 1+2+4+8+16+... is definitely an example of infinite...