If the Arithmetic mean and Geometrical mean of three numbers are equal to x, then the each number is equal to

Options

A. x
B. 2x
C. `sqrt(x)`
D. `3sqrt(x)`

Answer

To solve the problem, we need to find the values of the three numbers \( a, b, c \) given that their Arithmetic Mean (AM) and Geometric Mean (GM) are both equal to \( x \).

### Step-by-Step Solution:

1. **Understanding the Arithmetic Mean (AM)**:
The Arithmetic Mean of three numbers \( a, b, c \) is given by the formula:
\[ AM = \frac{a + b + c}{3} \]
According to the problem, this is equal to \( x \):
\[ \frac{a + b + c}{3} = x \]

2. **Rearranging the AM Equation**:
To eliminate the fraction, we can multiply both sides of the equation by 3:
\[ a + b + c = 3x \]

3. **Understanding the Geometric Mean (GM)**:
The Geometric Mean of the same three numbers \( a, b, c \) is given by the formula:
\[ GM = \sqrt[3]{abc} \]
According to the problem, this is also equal to \( x \):
\[ \sqrt[3]{abc} = x \]

4. **Cubing the GM Equation**:
To eliminate the cube root, we can cube both sides of the equation:
\[ abc = x^3 \]

5. **Analyzing the Two Equations**:
Now we have two equations:
- From the AM: \( a + b + c = 3x \)
- From the GM: \( abc = x^3 \)

6. **Finding the Values of \( a, b, c \)**:
The only way for both the AM and GM to be equal and for the equations to hold true is if all three numbers are equal. Let’s assume:
\[ a = b = c = k \]
Substituting this into the AM equation:
\[ k + k + k = 3k = 3x \implies k = x \]
Thus, each number \( a, b, c \) must be equal to \( x \).

### Conclusion:
Therefore, each number is equal to \( x \).

### Final Answer:
Each number is equal to \( x \).

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