If the `3^(rd)` term of a G.P. is 4, find the product of its first five terms.

Answer

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

### Step 1: Understand the terms of a G.P.
In a geometric progression, the terms can be expressed as:
- First term (T1) = a
- Second term (T2) = ar
- Third term (T3) = ar²
- Fourth term (T4) = ar³
- Fifth term (T5) = ar⁴

### Step 2: Use the information given
We know that the third term (T3) is 4:
\[ T3 = ar^2 = 4 \]

### Step 3: Find the product of the first five terms
The product of the first five terms can be expressed as:
\[ P = T1 \times T2 \times T3 \times T4 \times T5 \]
Substituting the terms:
\[ P = a \times ar \times ar^2 \times ar^3 \times ar^4 \]

### Step 4: Simplify the product
We can factor out the common terms:
\[ P = a^5 \times r^{0+1+2+3+4} \]
\[ P = a^5 \times r^{10} \]

### Step 5: Relate the product to the third term
Since we know that \( ar^2 = 4 \), we can express \( a \) in terms of \( r \):
\[ a = \frac{4}{r^2} \]

### Step 6: Substitute \( a \) in the product expression
Now substitute \( a \) into the product:
\[ P = \left(\frac{4}{r^2}\right)^5 \times r^{10} \]
\[ P = \frac{4^5}{r^{10}} \times r^{10} \]
\[ P = 4^5 \]

### Step 7: Calculate \( 4^5 \)
Now calculate \( 4^5 \):
\[ 4^5 = 1024 \]

### Final Answer
Thus, the product of the first five terms of the G.P. is:
\[ \boxed{1024} \]