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

Answer

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

### Step 1: Understand the formula for the nth term of a G.P.
The nth 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 \( a_8 = 192 \):
\[ a_8 = a \cdot r^{8-1} = a \cdot r^7 \]
Substituting the known values:
\[ a \cdot 2^7 = 192 \]

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

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

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

### Step 6: Write the equation for the 12th term.
Now we need to find the 12th term \( a_{12} \):
\[ a_{12} = a \cdot r^{12-1} = a \cdot r^{11} \]
Substituting the values of \( a \) and \( r \):
\[ a_{12} = \frac{3}{2} \cdot 2^{11} \]

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

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

### Step 9: Simplify to find \( a_{12} \).
Calculating \( a_{12} \):
\[ a_{12} = \frac{3 \cdot 2048}{2} = \frac{6144}{2} = 3072 \]

### Final Answer:
The 12th term of the G.P. is \( 3072 \).

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