If the product of 4 terms of a G.P. is 729 , then find the GM of the G.P.

Answer

To find the geometric mean (GM) of a geometric progression (G.P.) where the product of four terms is given as 729, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Terms of the G.P.**:
Let the first term of the G.P. be \( a \). The four terms can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)

2. **Write the Product of the Terms**:
The product of these four terms is given by:
\[ a \cdot ar \cdot ar^2 \cdot ar^3 = a^4 r^6 \]
According to the problem, this product equals 729:
\[ a^4 r^6 = 729 \]

3. **Express 729 in Terms of Powers**:
We can express 729 as a power of 3:
\[ 729 = 3^6 \]
Therefore, we have:
\[ a^4 r^6 = 3^6 \]

4. **Find the Geometric Mean**:
The geometric mean (GM) of the four terms can be calculated using the formula:
\[ \text{GM} = \sqrt[4]{a \cdot ar \cdot ar^2 \cdot ar^3} = \sqrt[4]{a^4 r^6} \]
This simplifies to:
\[ \text{GM} = \sqrt[4]{a^4} \cdot \sqrt[4]{r^6} = a \cdot r^{3/2} \]

5. **Relate GM to the Product**:
We know that:
\[ a^4 r^6 = 729 \implies a^4 = 729 / r^6 \]
Taking the fourth root:
\[ a = \sqrt[4]{729/r^6} \]

6. **Calculate the GM**:
To find the GM, we can also directly compute:
\[ \text{GM} = \sqrt[4]{729} = \sqrt[4]{3^6} = 3^{6/4} = 3^{3/2} = 3 \sqrt{3} \]

### Final Answer:
Thus, the geometric mean of the G.P. is \( 3\sqrt{3} \).

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