If the third term of G.P. is 4, then find the product of first five terms

Answer

To solve the problem step by step, we need to find the product of the first five terms of a geometric progression (G.P.) given that the third term is 4.

### Step-by-Step Solution:

1. **Understanding the G.P. Terms**:
The nth term of a G.P. can be expressed as:
\[ T_n = A \cdot r^{n-1} \]
where \( A \) is the first term and \( r \) is the common ratio.

2. **Finding the Third Term**:
We know that the third term \( T_3 \) is given as 4. Therefore, we can write:
\[ T_3 = A \cdot r^{3-1} = A \cdot r^2 \]
Setting this equal to 4 gives us:
\[ A \cdot r^2 = 4 \quad \text{(Equation 1)} \]

3. **Finding the Product of the First Five Terms**:
The first five terms of the G.P. are:
- First term: \( T_1 = A \)
- Second term: \( T_2 = A \cdot r \)
- Third term: \( T_3 = A \cdot r^2 \)
- Fourth term: \( T_4 = A \cdot r^3 \)
- Fifth term: \( T_5 = A \cdot r^4 \)

The product \( P \) of the first five terms is:
\[ P = T_1 \cdot T_2 \cdot T_3 \cdot T_4 \cdot T_5 = A \cdot (A \cdot r) \cdot (A \cdot r^2) \cdot (A \cdot r^3) \cdot (A \cdot r^4) \]

4. **Simplifying the Product**:
We can factor out \( A \) and \( r \):
\[ P = A^5 \cdot (r^0 \cdot r^1 \cdot r^2 \cdot r^3 \cdot r^4) = A^5 \cdot r^{0+1+2+3+4} = A^5 \cdot r^{10} \]

5. **Substituting \( A \cdot r^2 \)**:
From Equation 1, we know that \( A \cdot r^2 = 4 \). We can express \( A \) in terms of \( r \):
\[ A = \frac{4}{r^2} \]

6. **Substituting into the Product**:
Now substitute \( A \) into the product:
\[ P = \left(\frac{4}{r^2}\right)^5 \cdot r^{10} = \frac{4^5}{r^{10}} \cdot r^{10} = 4^5 \]

7. **Calculating \( 4^5 \)**:
Finally, we calculate \( 4^5 \):
\[ 4^5 = 1024 \]

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

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