Geometric Sequence,Means, and Infinite Geometric Sequence

Geometric Sequence

What is Geometric Sequence
-A sequence in which each term after the first term is obtained by multiplying.

Formula:
an=a1(r)

Example:
1.)4,8,16,32,...,a
12

a1=4 r=4
n=12 an=12

a
12=4(2)
a
12=4(2 )
a
12=2048=Final Answer

Infinite Geometric Sequence

Formula:
Sn=a
1(1-r )
1-r r=1

Sn=a
1-an
1-r r=1

Example:
3,6,12,24,...,S
10

n=10 r=2
a
1=3

S
10=3(1-2 )
1-2
=3(-1024)
-1
S
10=3069=Final Answer

Sum of an Infinite Geometric Sequence

Formula:
s=a
1
1-r

Divergent=|r|>1
Convergent=|r|<1

Example:
2, 6, 18,...
3 15 75

r=3 a
1=2
5 3

S= 2
3
1-3
5

S=5
3

Geometric Means

Formula:
an=a1(r)

Example:
1.)2,10,50,250,1250

r=? n=5
a
n=1250 a1=2

Sn=1250=2r
1250=2r
1250=2r
2 2

625=r

5=r=Final Answer

Labels:

a1=First Term

an=The nth term

n=The Number of Terms

r=Common Ratio

d=Distance