The arithmetic mean, geometric mean and median of six positive numbers a, a, b, b, c, c where a

Question

The arithmetic mean, geometric mean and median of six positive numbers a, a, b, b, c, c where a < b < c are $\frac{7}{3}$, 2, 2 respectively. Then what is the sum of the squares of all the six numbers ?

Accepted Answer

To solve the problem, we need to analyze the given conditions and verify why Option 2 is the correct answer.

We have six positive numbers: $a, a, b, b, c, c$ where $a < b < c$. The conditions given are:

1. The arithmetic mean is $\frac{7}{3}$.
2. The geometric mean is 2.
3. The median is 2.

Let's break down each condition:

### Arithmetic Mean
The arithmetic mean of the numbers is given by:
$$
\frac{a + a + b + b + c + c}{6} = \frac{7}{3}
$$
Simplifying, we have:
$$
\frac{2a + 2b + 2c}{6} = \frac{7}{3}
$$
$$
\frac{a + b + c}{3} = \frac{7}{3}
$$
$$
a + b + c = 7
$$

### Geometric Mean
The geometric mean is given by:
$$
(a^2 \cdot b^2 \cdot c^2)^{1/6} = 2
$$
$$
(a \cdot b \cdot c)^{1/3} = 2
$$
$$
a \cdot b \cdot c = 8
$$

### Median
The median of the numbers is 2. Since the numbers are $a, a, b, b, c, c$ and $a < b < c$, the median is the average of the two middle numbers, which are both $b$. Therefore, $b = 2$.

### Solving the Equations
Now, we have:
1. $a + 2 + c = 7$ (since $b = 2$)
2. $a \cdot 2 \cdot c = 8$

From the first equation:
$$
a + c = 5
$$

Substitute $c = 5 - a$ into the second equation:
$$
a \cdot 2 \cdot (5 - a) = 8
$$
$$
2a(5 - a) = 8
$$
$$
10a - 2a^2 = 8
$$
$$
2a^2 - 10a + 8 = 0
$$
Divide the entire equation by 2:
$$
a^2 - 5a + 4 = 0
$$
Factoring the quadratic equation:
$$
(a - 4)(a - 1) = 0
$$
Thus, $a = 4$ or $a = 1$.

Since $a < b < c$ and $b = 2$, $a$ must be 1. Therefore, $c = 5 - 1 = 4$.

### Sum of Squares
Now, we calculate the sum of the squares of all six numbers:
$$
a^2 + a^2 + b^2 + b^2 + c^2 + c^2 = 1^2 + 1^2 + 2^2 + 2^2 + 4^2 + 4^2
$$
$$
= 1 + 1 + 4 + 4 + 16 + 16
$$
$$
= 42
$$

Thus, the sum of the squares of all the six numbers is 42, which corresponds to Option 2.

### Explanation of Incorrect Options
- **Option 1 (40):** This sum would result from incorrect calculations or assumptions about the values of $a$, $b$, or $c$.
- **Option 3 (45):** This sum does not satisfy the given conditions when recalculating the squares.
- **Option 4 (48):** This sum exceeds the correct calculation based on the given conditions.

Therefore, Option 2 is the correct answer, as it aligns with the conditions provided in the problem.