The fifth term of a G.P. is 32 and common ratio is
2 , then the sum of first 14 terms of the G.P. is

Options

A. 16388
B. 32667
C. 32766
D. 64342

Answer

To solve the problem step by step, we will follow the information provided and apply the formulas related to geometric progressions (G.P.).

### Step 1: Identify the given information
We know that:
- The fifth term of the G.P. (T5) is 32.
- The common ratio (r) is 2.

### Step 2: Write the formula for the nth term of a G.P.
The nth term of a G.P. is given by the formula:
\[ T_n = a \cdot r^{n-1} \]
where:
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

### Step 3: Apply the formula to find the fifth term
For the fifth term (n = 5):
\[ T_5 = a \cdot r^{5-1} = a \cdot r^4 \]
Given that \( T_5 = 32 \), we can write:
\[ a \cdot r^4 = 32 \]

### Step 4: Substitute the value of r
Since the common ratio \( r = 2 \):
\[ a \cdot 2^4 = 32 \]
Calculating \( 2^4 \):
\[ 2^4 = 16 \]
So, we have:
\[ a \cdot 16 = 32 \]

### Step 5: Solve for a
To find \( a \):
\[ a = \frac{32}{16} = 2 \]

### Step 6: Write the formula for the sum of the first n terms of a G.P.
The sum of the first n terms (S_n) of a G.P. is given by:
\[ S_n = a \cdot \frac{r^n - 1}{r - 1} \]
where \( n \) is the number of terms.

### Step 7: Substitute the values to find the sum of the first 14 terms
We need to find \( S_{14} \):
- \( a = 2 \)
- \( r = 2 \)
- \( n = 14 \)

Substituting these values into the sum formula:
\[ S_{14} = 2 \cdot \frac{2^{14} - 1}{2 - 1} \]
Since \( 2 - 1 = 1 \), this simplifies to:
\[ S_{14} = 2 \cdot (2^{14} - 1) \]

### Step 8: Calculate \( 2^{14} \)
Calculating \( 2^{14} \):
\[ 2^{14} = 16384 \]
So:
\[ S_{14} = 2 \cdot (16384 - 1) = 2 \cdot 16383 \]

### Step 9: Final calculation
Now, calculate:
\[ S_{14} = 32766 \]

### Conclusion
The sum of the first 14 terms of the G.P. is:
\[ \boxed{32766} \]