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Answer:
Each term in the sequence adds another digit '1' to the previous term:
1. First term: ( 0.1 )
2. Second term: ( 0.11 )
3. Third term: ( 0.111 )
4. Fourth term: ( 0.1111 )
We can express each term as:
[tex]a_n = 0.\underbrace{111\ldots1}_{n \text{ ones}}[/tex]
To write this in a more mathematical form, we can use the fact that each term is a sum of fractions:
[tex]a_n = \sum_{k=1}^{n} \frac{1}{10^k}[/tex]
Alternatively, we can represent each term as a finite geometric series:
[tex]a_n = \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \cdots + \frac{1}{10^n}[/tex]
This geometric series can be simplified using the formula for the sum of a geometric series:
[tex]a_n = \frac{1 - \left(\frac{1}{10}\right)^n}{10 - 1}[/tex]
Since ( 10 - 1 = 9 ):
[tex]a_n = \frac{1 - 10^{-n}}{9}[/tex]
So the rule for the nth term of the sequence is:
[tex]a_n = \frac{1 - 10^{-n}}{9}[/tex]