The third term of a G.P. is 4. The product of the first five terms is

Options

A. `4^(3)`
B. `4^(5)`
C. `4^(4)`
D. none of these

Answer

To solve the problem step by step, we will denote the first term of the geometric progression (G.P.) as \( A \) and the common ratio as \( r \).

### Step 1: Identify the third term of the G.P.
The third term of a G.P. is given by the formula:
\[ T_3 = A r^2 \]
According to the problem, the third term is 4. Therefore, we have:
\[ A r^2 = 4 \quad \text{(1)} \]

### Step 2: Write the first five terms of the G.P.
The first five terms of the G.P. are:
- First term: \( T_1 = A \)
- Second term: \( T_2 = A r \)
- Third term: \( T_3 = A r^2 \)
- Fourth term: \( T_4 = A r^3 \)
- Fifth term: \( T_5 = A r^4 \)

### Step 3: Calculate the product of the first five terms.
The product of the first five terms is given by:
\[ P = T_1 \times T_2 \times T_3 \times T_4 \times T_5 \]
Substituting the expressions for the terms, we have:
\[ P = A \times (A r) \times (A r^2) \times (A r^3) \times (A r^4) \]
This simplifies to:
\[ P = A^5 \times r^{0 + 1 + 2 + 3 + 4} \]
Calculating the exponent of \( r \):
\[ 0 + 1 + 2 + 3 + 4 = 10 \]
Thus, we can express the product as:
\[ P = A^5 \times r^{10} \quad \text{(2)} \]

### Step 4: Substitute \( A r^2 \) from equation (1) into equation (2).
From equation (1), we have \( A r^2 = 4 \). We can express \( A \) in terms of \( r \):
\[ A = \frac{4}{r^2} \]
Now substitute this into the product equation:
\[ P = \left(\frac{4}{r^2}\right)^5 \times r^{10} \]
Calculating \( \left(\frac{4}{r^2}\right)^5 \):
\[ P = \frac{4^5}{r^{10}} \times r^{10} \]
The \( r^{10} \) terms cancel out:
\[ P = 4^5 \]

### Step 5: Calculate \( 4^5 \).
Calculating \( 4^5 \):
\[ 4^5 = 1024 \]

### Final Answer:
The product of the first five terms of the G.P. is \( 1024 \).

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