The first three terms of a geometric sequence are given: $$ \frac{1}{3} ;-1 ; 3 ; \ldots $$ 1. Find the constant ratio, $$ r $$. 2. Determine the formula for $$ T_{n} $$ of the sequence.

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Let's analyze the given geometric sequence and determine the required values step by step.

## Given Sequence:
$$ \frac{1}{3};\ -1;\ 3; \ldots $$

## 1. Finding the Constant Ratio, $$ r $$:

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio $$ r $$. Mathematically, this can be expressed as:
$$ r = \frac{T_{n}}{T_{n-1}} $$

Let's calculate $$ r $$ using the first and second terms:
$$
r = \frac{T_2}{T_1} = \frac{-1}{\frac{1}{3}} = -1 \times 3 = -3
$$

**Therefore, the constant ratio is $$ r = -3 $$.**

## 2. Determining the Formula for $$ T_{n} $$:

The general formula for the $$ n $$-th term of a geometric sequence is:
$$
T_n = T_1 \times r^{(n-1)}
$$
where:
- $$ T_n $$ is the $$ n $$-th term,
- $$ T_1 $$ is the first term,
- $$ r $$ is the common ratio,
- $$ n $$ is the term number.

Using the values from the given sequence:
- $$ T_1 = \frac{1}{3} $$
- $$ r = -3 $$

Substitute these into the general formula:
$$
T_n = \frac{1}{3} \times (-3)^{(n-1)}
$$

**Therefore, the formula for the $$ n $$-th term is:**
$$
T_n = \frac{1}{3} \times (-3)^{n-1}
$$

You can also express this formula in alternative forms, such as:
$$
T_n = \frac{(-3)^{n-1}}{3} \quad \text{or} \quad T_n = (-3)^{n-1} \times \frac{1}{3}
$$
Both representations are mathematically equivalent and valid.

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**Summary:**
1. **Constant ratio, $$ r = -3 $$.**
2. **General term formula, $$ T_n = \frac{1}{3} \times (-3)^{n-1} $$.**
The constant ratio $$ r $$ is -3. The formula for the $$ n $$-th term is $$ T_n = \frac{1}{3} \times (-3)^{n-1} $$.

The first three terms of a geometric sequence are given: \[ \frac{1}{3} ;-1 ; 3 ; \ldots \] 1. Find the constant ratio, \( r \). 2. Determine the formula for \( T_{n} \) of the sequence.

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