The first three terms of a geometric progression are 3, -1 and `1/3`. The next term of the progression is

Options

A. 2
B. `-2`
C. `(-1)/(9)`
D. `(-5)/(9)`

Answer

To find the next term in the geometric progression (GP) given the first three terms 3, -1, and \( \frac{1}{3} \), we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \( a \) of the GP is given as:
\[ a = 3 \]

### Step 2: Calculate the common ratio \( r \)
The common ratio \( r \) can be found by dividing the second term by the first term:
\[ r = \frac{\text{second term}}{\text{first term}} = \frac{-1}{3} \]

### Step 3: Verify the common ratio with the third term
To ensure that the ratio is consistent, we can also check the ratio between the third term and the second term:
\[ r = \frac{\text{third term}}{\text{second term}} = \frac{\frac{1}{3}}{-1} = -\frac{1}{3} \]
This confirms that the common ratio is indeed \( r = -\frac{1}{3} \).

### Step 4: Calculate the fourth term
The formula for the \( n \)-th term of a GP is given by:
\[ a_n = a \cdot r^{n-1} \]
To find the fourth term (\( n = 4 \)):
\[ a_4 = 3 \cdot \left(-\frac{1}{3}\right)^{4-1} \]
\[ a_4 = 3 \cdot \left(-\frac{1}{3}\right)^{3} \]

### Step 5: Simplify the expression
Calculating \( \left(-\frac{1}{3}\right)^{3} \):
\[ \left(-\frac{1}{3}\right)^{3} = -\frac{1}{27} \]
Now substitute this back into the equation for \( a_4 \):
\[ a_4 = 3 \cdot \left(-\frac{1}{27}\right) \]
\[ a_4 = -\frac{3}{27} \]
\[ a_4 = -\frac{1}{9} \]

### Conclusion
The next term of the progression is:
\[ \boxed{-\frac{1}{9}} \]