## The Curious Case of the 1 1/3 1/9 Sequence

The sequence 1, 1/3, 1/9... might seem simple at first glance, but it hides some interesting mathematical properties and connections to other concepts. Let's explore this sequence and uncover its secrets.

### Recognizing the Pattern

The most obvious characteristic of this sequence is its **geometric progression**. Each term is obtained by multiplying the previous term by 1/3. This constant ratio is what defines a geometric sequence.

### Finding the nth Term

To find the nth term of this sequence, we can use the formula for geometric sequences:

**a<sub>n</sub> = a<sub>1</sub> * r<sup>n-1</sup>**

Where:

**a<sub>n</sub>**is the nth term**a<sub>1</sub>**is the first term (1 in this case)**r**is the common ratio (1/3 in this case)**n**is the term number

Therefore, the nth term of this sequence is: **a<sub>n</sub> = 1 * (1/3)<sup>n-1</sup> = (1/3)<sup>n-1</sup>**

### Infinite Geometric Series

This sequence is also closely related to **infinite geometric series**. An infinite geometric series is the sum of an infinite number of terms in a geometric sequence. For this specific sequence, the infinite geometric series is:

**1 + 1/3 + 1/9 + 1/27 + ...**

This series has a special property: **it converges to a finite value**. This is because the common ratio (1/3) is less than 1. The sum of this infinite geometric series is:

**S = a<sub>1</sub> / (1 - r) = 1 / (1 - 1/3) = 3/2**

### Applications

This sequence and its associated infinite geometric series have various applications in different fields:

**Physics:**This sequence can model the decay of radioactive substances.**Finance:**It can be used to calculate the present value of future cash flows in discounted cash flow analysis.**Computer science:**The concept of geometric series is fundamental to understanding the complexity of algorithms.

### In Conclusion

The seemingly simple 1, 1/3, 1/9... sequence reveals fascinating mathematical connections and has practical applications in various fields. Its geometric progression, convergence to a finite value in its infinite series, and its relation to other mathematical concepts make it a captivating example of the power and beauty of mathematics.