In a G.P first term is `3/4`, common ratio is 2 and the last term is 384 , the number of terms of G.P. is (i) 8 (ii) 9 (iii) 10 (iv) 11

Options

A. 8
B. 9
C. 10
D. 11

Answer

To find the number of terms in the given geometric progression (G.P.), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- First term \( a = \frac{3}{4} \)
- Common ratio \( r = 2 \)
- Last term \( a_n = 384 \)

2. **Use the Formula for the nth Term of a G.P.**:
The formula for the nth term of a G.P. is given by:
\[ a_n = a \cdot r^{n-1} \]
Substituting the known values into the formula:
\[ 384 = \frac{3}{4} \cdot 2^{n-1} \]

3. **Isolate the Exponential Term**:
To isolate \( 2^{n-1} \), multiply both sides by \( \frac{4}{3} \):
\[ 384 \cdot \frac{4}{3} = 2^{n-1} \]
Calculate the left side:
\[ 384 \cdot \frac{4}{3} = \frac{1536}{3} = 512 \]
Thus, we have:
\[ 2^{n-1} = 512 \]

4. **Express 512 as a Power of 2**:
We know that:
\[ 512 = 2^9 \]
Therefore, we can equate the powers:
\[ n - 1 = 9 \]

5. **Solve for n**:
Adding 1 to both sides gives:
\[ n = 10 \]

6. **Conclusion**:
The number of terms in the G.P. is \( n = 10 \).

### Final Answer:
The number of terms of the G.P. is **10** (option iii).