The fourth term of a G.P. is 4. Find product of its first seven terms.

Answer

To find the product of the first seven terms of a geometric progression (G.P.) given that the fourth term is 4, we can follow these steps:

### Step 1: Define the terms of the G.P.
Let the first term of the G.P. be \( a \) and the common ratio be \( r \). The terms of the G.P. can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)
- Fifth term: \( ar^4 \)
- Sixth term: \( ar^5 \)
- Seventh term: \( ar^6 \)

### Step 2: Use the information about the fourth term
We know that the fourth term is given as 4:
\[ ar^3 = 4 \]

### Step 3: Write the product of the first seven terms
The product \( P \) of the first seven terms can be expressed as:
\[ P = a \cdot ar \cdot ar^2 \cdot ar^3 \cdot ar^4 \cdot ar^5 \cdot ar^6 \]
This simplifies to:
\[ P = a^7 \cdot r^{0 + 1 + 2 + 3 + 4 + 5 + 6} \]
The sum of the exponents of \( r \) is:
\[ 0 + 1 + 2 + 3 + 4 + 5 + 6 = 21 \]
Thus, we can write:
\[ P = a^7 \cdot r^{21} \]

### Step 4: Express \( a \) in terms of \( r \)
From the equation \( ar^3 = 4 \), we can express \( a \) as:
\[ a = \frac{4}{r^3} \]

### Step 5: Substitute \( a \) back into the product
Now substitute \( a \) into the product \( P \):
\[ P = \left(\frac{4}{r^3}\right)^7 \cdot r^{21} \]
This simplifies to:
\[ P = \frac{4^7}{r^{21}} \cdot r^{21} \]
The \( r^{21} \) terms cancel out:
\[ P = 4^7 \]

### Step 6: Calculate \( 4^7 \)
Now, we calculate \( 4^7 \):
\[ 4^7 = (2^2)^7 = 2^{14} = 16384 \]

### Final Answer
Thus, the product of the first seven terms of the G.P. is:
\[ \boxed{16384} \]