The next number in pattern 1/324, 1/108, 1/36, 1/12 would be 1/4
Further explanation
Firstly , let us learn about types of sequence in mathematics.
Arithmetic Progression is a sequence of numbers in which each of adjacent numbers have a constant difference.
[tex]\boxed{T_n = a + (n-1)d}[/tex]
[tex]\boxed{S_n = \frac{1}{2}n ( 2a + (n-1)d )}[/tex]
Tn = n-th term of the sequence
Sn = sum of the first n numbers of the sequence
a = the initial term of the sequence
d = common difference between adjacent numbers
Geometric Progression is a sequence of numbers in which each of adjacent numbers have a constant ration.
[tex]\boxed{T_n = a ~ r^{n-1}}[/tex]
[tex]\boxed{S_n = \frac{a( 1 - r^n ) }{1 - r}}[/tex]
Tn = n-th term of the sequence
Sn = sum of the first n numbers of the sequence
a = the initial term of the sequence
r = common ratio between adjacent numbers
Let us now tackle the problem!
Given:
This problem is about Geometric Sequence. Let's see why.
[tex]\frac{1}{324} , \frac{1}{108} , \frac{1}{36} , \frac{1}{12} , . . .[/tex]
T₁ = [tex]\frac{1}{324}[/tex]
T₂ = [tex]\frac{1}{108}[/tex]
T₃ = [tex]\frac{1}{36}[/tex]
T₄ = [tex]\frac{1}{12}[/tex]
[tex]\large {\boxed {\frac{T_2}{T_1} = \frac{T_3}{T_2} = \frac{T_4}{T_3} = r = 3} }[/tex]
From the above results it can be concluded that this is Geometric Sequence.
The next number will be the fifth term and can be calculated as following.
[tex]T_n = a ~ r^{n-1}[/tex]
[tex]T_5 = \frac{1}{324} \times 3^{5 - 1}[/tex]
[tex]T_5 = \frac{1}{324} \times 3^4[/tex]
[tex]T_5 = \frac{1}{324} \times 81[/tex]
[tex]\large {\boxed {T_5 = \frac{1}{4} } }[/tex]
Learn more
- Geometric Series : https://brainly.com/question/4520950
- Arithmetic Progression : https://brainly.com/question/2966265
- Geometric Sequence : https://brainly.com/question/2166405
Answer details
Grade: Middle School
Subject: Mathematics
Chapter: Arithmetic and Geometric Series
Keywords: Arithmetic , Geometric , Series , Sequence , Difference , Term
