Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.

Explanation

To find the 12th term of a geometric progression (G.P.) where the 8th term is 192 and the common ratio is 2, we can follow these steps:

### Step 1: Understand the formula for the n-th term of a G.P.
The n-th term of a G.P. can be expressed as:
\[ A_n = A \cdot R^{n-1} \]
where:
- \( A \) is the first term,
- \( R \) is the common ratio,
- \( n \) is the term number.

### Step 2: Write the equation for the 8th term.
Given that the 8th term \( A_8 = 192 \) and the common ratio \( R = 2 \), we can write:
\[ A_8 = A \cdot R^{8-1} \]
Substituting the values we have:
\[ 192 = A \cdot 2^7 \]

### Step 3: Calculate \( 2^7 \).
Calculate \( 2^7 \):
\[ 2^7 = 128 \]

### Step 4: Substitute \( 2^7 \) into the equation.
Now substituting \( 128 \) into the equation:
\[ 192 = A \cdot 128 \]

### Step 5: Solve for \( A \).
To find \( A \), divide both sides by \( 128 \):
\[ A = \frac{192}{128} \]
Simplifying this gives:
\[ A = \frac{3}{2} \]

### Step 6: Write the equation for the 12th term.
Now we need to find the 12th term \( A_{12} \):
\[ A_{12} = A \cdot R^{12-1} \]
Substituting the known values:
\[ A_{12} = \frac{3}{2} \cdot 2^{11} \]

### Step 7: Calculate \( 2^{11} \).
Calculate \( 2^{11} \):
\[ 2^{11} = 2048 \]

### Step 8: Substitute \( 2^{11} \) into the equation for \( A_{12} \).
Now substituting \( 2048 \) into the equation:
\[ A_{12} = \frac{3}{2} \cdot 2048 \]

### Step 9: Simplify to find \( A_{12} \).
Calculating this gives:
\[ A_{12} = \frac{3 \cdot 2048}{2} = 3 \cdot 1024 = 3072 \]

### Final Answer:
Thus, the 12th term of the G.P. is:
\[ \boxed{3072} \]

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