IDNStudy.com, ang iyong destinasyon para sa malinaw at mabilis na mga sagot. Makakuha ng impormasyon mula sa aming mga eksperto, na nagbibigay ng detalyadong sagot sa lahat ng iyong mga tanong.
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]