Answer :
Let's look at the sequences given in the question and analyze their patterns one by one:
a) 2013, 9, 27, 81, 243
This sequence doesn't use a simple subtraction of the previous term by 3. Instead, it appears to be a geometric sequence where each term is obtained by multiplying the previous term by 3. The sequence can be represented as:
- First term: [tex]3^1 = 3[/tex]
- Second term: [tex]3^2 = 9[/tex]
- Third term: [tex]3^3 = 27[/tex]
- Fourth term: [tex]3^4 = 81[/tex]
- Fifth term: [tex]3^5 = 243[/tex]
Each term is the previous term multiplied by 3, hence it's a geometric sequence with a common ratio of 3.
b) 5, 25, 125, 625, 3125
This sequence is also geometric, with each term being 5 times the previous term. It can be expressed as:
- First term: [tex]5^1 = 5[/tex]
- Second term: [tex]5^2 = 25[/tex]
- Third term: [tex]5^3 = 125[/tex]
- Fourth term: [tex]5^4 = 625[/tex]
- Fifth term: [tex]5^5 = 3125[/tex]
The common ratio here is 5.
c) 128, 64, 32, 16, 8, 4, 2, 1
This is also a geometric sequence where each term is obtained by dividing the previous term by 2. It can be described as:
- First term: [tex]2^7 = 128[/tex]
- Next, dividing by 2 repeatedly gives the subsequent terms as: 64, 32, 16, 8, 4, 2, 1.
The common ratio is [tex]\frac{1}{2}[/tex].
d) 100000, 10000, 1000, 100, 10
This sequence appears to be a geometric sequence where each term is obtained by dividing the previous term by 10:
- First term: [tex]10^5 = 100000[/tex]
- Second term: [tex]10^4 = 10000[/tex]
- Third term: [tex]10^3 = 1000[/tex]
- Fourth term: [tex]10^2 = 100[/tex]
- Fifth term: [tex]10^1 = 10[/tex]
The common ratio is [tex]\frac{1}{10}[/tex].
In summary, each of these sequences demonstrates a pattern where the terms progress in a consistent geometric manner, either by multiplying or dividing by a fixed number (common ratio).
