In the previous discussion under the title: :Arithmetic Problem Solving, we have tried to understand the sequence and the arithmetic series with some sample questions. Continued tutorial we try times about sequences and geometric series.
Rows and Geometry Rows
First we will understand the initial concept or the basics of the geometry sequence which includes:- What is a sequence of geometry ?
- What is a geometry series ?
What is a Geometric Row?
The geometric row is a row that has a fixed ratio between two consecutive sequential rows. If in the arithmetic sequence, the difference between one term and the next is called a different value. While in the geometry sequence the difference between terms termed by ratio ( denoted by r).Suppose the line is known as below:
The sequence of numbers has a fixed ratio of 3 or r = 3. That means the sequence is a sequence of geometries.
Another Example of Row Geometry:
| 2, 4, 8, 16, 32, 64, 128, 256, ... |
Each term (except the first term) is the product of the previous by 2.
In general we can write the Geometric Sequence (Order) as follows:
{a, ar, ar2, ar3, ar4, ar5, ar6, ar7...}
where:
- a is the first term
- r is the ratio
Geometry Row Formulas
1. To find the nth Tribe :
Un = ar(n-1)
where :
where :
- Un is the nth term
- a denotes the first term
- r represents the ratio
- n states the number of tribes
2. To find the ratio value (r):
r = Un U(n-1)
where :
where :
- r ia the ratio
- Un is the nth term
- U(n-1) is the previous nth term
3. Finding the Middle Tribe
We can find the middle tribe for a geometric sequence that has an odd number (the number of tribes must be odd) where the first term and ratio are known, then the formula is used:
Ut = √a . rn
where:
- Ut is the middle term
- a is the first term
- n states the number of tribes
- r is the ratio
But if to find the middle tribe whose condition is only known to the first tribe, the number of n tribes and the last tribe, then the formula:
Ut = √a . Un
where :
- Ut is the middle term
- a is the first term
- Un is the nth term (in this case as the last term)
What is Geometry Series ?
Just as the arithmetic series is the sum of the arithmetic sequences, the geometry series is the sum of the tribal values of a geometric sequence. Geometric sequences are also known as geometric progression.
Example:
Example:
- 1 + 2 + 4 + 8 +16+32
- 3 + 6 + 12 + 24 + 48 + 96
To calculate the geometry series there are two formulas, namely:
- Geometry Derivation
Formulas Descending The derived geometry sequence can only be used if 0 < r < 1Sn = a(1 - rn) 1 - r
where:
- Sn is the sum of a series of nth terms
- a is the first term
- r is the ratio
- n is the number of tribes
- Geometry Derivation
Formulas Up The rising geometry sequence can only be used if r > 1.Sn = a(rn-1) r - 1
where :
- Sn is the sum of a series of nth terms
- a is the first term
- r is the ratio
- n is the number of tribes
Exercises
Problem No.1Given a sequence of geometries 4, 8, 16 .... then the sixth of the sequence of geometries:
a. 128
b. 168
c. 68
d. 180
Discussion
a = 4
r = 2
Un = ar(n-1)
⇒ 4.2(6-1)
⇒ 4.2(6-1)
⇒ 4.2(5)
⇒ 128
r = 2
Un = ar(n-1)
⇒ 4.2(6-1)
⇒ 4.2(6-1)
⇒ 4.2(5)
⇒ 128
Answer : a
Problem No.2
Given a sequence of geometries: 3, 9, 27, 81, 243. What is the ratio of the geometry sequence:
a. 4
b. 3
c. 2
d. 9
Discussion
We take the last two numbers: 81 and 243, then:
Un = 243
U(n-1) = 81
So the ratio value (r) :
r = Un U(n-1) = 243 81= 3
Un = 243
U(n-1) = 81
So the ratio value (r) :
r = Un U(n-1) = 243 81= 3
Answer :b
Problem No.3
Given a sequence of geometries: 5, 10, 20, 40, 80, ...., 5120. The middle term is:
a. 160
b. 320
c. 510
d. 640
Discussion
a = 5
Un = 5120
Un = 5120
Ut = √a . Un
Ut = √5 . 5120= √25600 = 160
Answer :a
Problem No.4
There is a sequence of geometries of five tribes. If the first term is 3 and the ratio is 3. What is the middle term?
a. 27
b. 81
c. 243
d. 9
Discussion
a = 3
r = 3
n = 5
r = 3
n = 5
Ut = √a . rn= √3 . 35=729 = 27
Answer : a