Find the G.M. of 3,6,12,24, 48 .

Answer

To find the Geometric Mean (G.M.) of the numbers 3, 6, 12, 24, and 48, we can follow these steps:

### Step 1: Identify the numbers
The numbers we need to find the G.M. for are:
- \( a_1 = 3 \)
- \( a_2 = 6 \)
- \( a_3 = 12 \)
- \( a_4 = 24 \)
- \( a_5 = 48 \)

### Step 2: Count the number of terms
We have a total of 5 terms (n = 5).

### Step 3: Calculate the product of the terms
Now, we need to calculate the product of these terms:
\[ \text{Product} = a_1 \times a_2 \times a_3 \times a_4 \times a_5 = 3 \times 6 \times 12 \times 24 \times 48 \]

### Step 4: Calculate the product step-by-step
Let's calculate the product step-by-step:
- First, calculate \( 3 \times 6 = 18 \)
- Next, calculate \( 18 \times 12 = 216 \)
- Then, calculate \( 216 \times 24 = 5184 \)
- Finally, calculate \( 5184 \times 48 = 248832 \)

So, the product of the terms is \( 248832 \).

### Step 5: Apply the formula for Geometric Mean
The formula for the Geometric Mean (G.M.) of n numbers is given by:
\[ \text{G.M.} = \sqrt[n]{\text{Product of the terms}} \]
In this case, we need to find the 5th root of the product:
\[ \text{G.M.} = \sqrt[5]{248832} \]

### Step 6: Calculate the 5th root
To find the 5th root of \( 248832 \), we can simplify:
\[ \sqrt[5]{248832} = 12 \]

### Conclusion
Thus, the Geometric Mean of the numbers 3, 6, 12, 24, and 48 is:
\[ \text{G.M.} = 12 \]

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