Find the next three terms of the series :
`(2)/(27),(2)/(9),(2)/(3), . . . . . . . . . . .`

Answer

To find the next three terms of the series \( \frac{2}{27}, \frac{2}{9}, \frac{2}{3}, \ldots \), we will follow these steps:

### Step 1: Identify the Series Type
We need to determine if the series is an Arithmetic Progression (AP) or a Geometric Progression (GP).

### Step 2: Calculate the Common Ratio
To check if the series is a GP, we will calculate the common ratio \( r \) using the formula:
\[ r = \frac{t_n}{t_{n-1}} \]
where \( t_n \) is the nth term and \( t_{n-1} \) is the previous term.

### Step 3: Calculate \( r \) for the First Two Terms
Let’s calculate \( r \) for the first two terms:
\[ t_1 = \frac{2}{27}, \quad t_2 = \frac{2}{9} \]
\[ r = \frac{t_2}{t_1} = \frac{\frac{2}{9}}{\frac{2}{27}} = \frac{2}{9} \times \frac{27}{2} = \frac{27}{9} = 3 \]

### Step 4: Calculate \( r \) for the Next Two Terms
Now, let’s calculate \( r \) for the second and third terms:
\[ t_2 = \frac{2}{9}, \quad t_3 = \frac{2}{3} \]
\[ r = \frac{t_3}{t_2} = \frac{\frac{2}{3}}{\frac{2}{9}} = \frac{2}{3} \times \frac{9}{2} = \frac{9}{3} = 3 \]

### Step 5: Confirm the Series is a GP
Since both calculations yield the same common ratio \( r = 3 \), we confirm that the series is a Geometric Progression (GP).

### Step 6: Find the Next Three Terms
To find the next terms \( t_4, t_5, t_6 \), we will use the common ratio \( r \):
- For \( t_4 \):
\[ t_4 = r \times t_3 = 3 \times \frac{2}{3} = 2 \]
- For \( t_5 \):
\[ t_5 = r \times t_4 = 3 \times 2 = 6 \]
- For \( t_6 \):
\[ t_6 = r \times t_5 = 3 \times 6 = 18 \]

### Final Answer
The next three terms of the series are:
\[ 2, 6, 18 \]

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