Find the 6th term of the progression 2, 6, 18,….

Explanation

To find the 6th term of the progression 2, 6, 18, ..., we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Type of Progression**:
The given sequence is 2, 6, 18. We need to determine if this is a geometric progression (GP).

2. **Calculate the Common Ratio**:
To check if it's a GP, we calculate the common ratio (r):
- The ratio between the second term and the first term:
\[ r = \frac{6}{2} = 3 \]
- The ratio between the third term and the second term:
\[ r = \frac{18}{6} = 3 \]
Since both ratios are equal, the sequence is a geometric progression with a common ratio \( r = 3 \).

3. **Identify the First Term**:
The first term \( a \) of the progression is:
\[ a = 2 \]

4. **Use the Formula for the nth Term of a GP**:
The formula for the nth term \( T_n \) of a geometric progression is given by:
\[ T_n = a \cdot r^{(n-1)} \]
We need to find the 6th term (\( n = 6 \)).

5. **Substitute the Values into the Formula**:
Substitute \( a = 2 \), \( r = 3 \), and \( n = 6 \) into the formula:
\[ T_6 = 2 \cdot 3^{(6-1)} = 2 \cdot 3^5 \]

6. **Calculate \( 3^5 \)**:
First, calculate \( 3^5 \):
\[ 3^5 = 243 \]

7. **Calculate \( T_6 \)**:
Now substitute back to find \( T_6 \):
\[ T_6 = 2 \cdot 243 = 486 \]

### Final Answer:
The 6th term of the progression is \( 486 \).

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