`((1.2.4+2.4.8+3.6.12+.....)/(1.3.9+2.6.18+3.9.27+.....))^(1//3)=?`

Options

A. `1/3`
B. `2/3`
C. `3/4`
D. `1/2`

Answer

To solve the given problem, we need to evaluate the expression:

\[ \left( \frac{1 \cdot 2 \cdot 4 + 2 \cdot 4 \cdot 8 + 3 \cdot 6 \cdot 12 + \ldots}{1 \cdot 3 \cdot 9 + 2 \cdot 6 \cdot 18 + 3 \cdot 9 \cdot 27 + \ldots} \right)^{\frac{1}{3}} \]

### Step 1: Analyze the Numerator

The numerator is:

\[ 1 \cdot 2 \cdot 4 + 2 \cdot 4 \cdot 8 + 3 \cdot 6 \cdot 12 + \ldots \]

We can factor out common terms. Notice that:

- The first term \(1 \cdot 2 \cdot 4\) can be written as \(1 \cdot 2 \cdot 4\).
- The second term \(2 \cdot 4 \cdot 8\) can be factored as \(2 \cdot 4 \cdot (2 \cdot 4)\).
- The third term \(3 \cdot 6 \cdot 12\) can be factored as \(3 \cdot 6 \cdot (2 \cdot 6)\).

We can factor out \(4\) from the first term, \(8\) from the second term, and \(12\) from the third term.

This gives us:

\[ = 4(1 + 2 \cdot 2 + 3 \cdot 2 + \ldots) \]

### Step 2: Identify the Pattern in the Numerator

The series \(1 + 2 \cdot 2 + 3 \cdot 2 + \ldots\) can be expressed as:

\[ = 1 + 2^2 + 3^2 + \ldots \]

This is an infinite series that can be summed up.

### Step 3: Analyze the Denominator

The denominator is:

\[ 1 \cdot 3 \cdot 9 + 2 \cdot 6 \cdot 18 + 3 \cdot 9 \cdot 27 + \ldots \]

Similar to the numerator, we can factor out common terms. Notice that:

- The first term \(1 \cdot 3 \cdot 9\) can be written as \(1 \cdot 3 \cdot 9\).
- The second term \(2 \cdot 6 \cdot 18\) can be factored as \(2 \cdot 6 \cdot (3 \cdot 6)\).
- The third term \(3 \cdot 9 \cdot 27\) can be factored as \(3 \cdot 9 \cdot (3 \cdot 9)\).

This gives us:

\[ = 9(1 + 2 \cdot 2 + 3 \cdot 3 + \ldots) \]

### Step 4: Identify the Pattern in the Denominator

The series \(1 + 2 \cdot 2 + 3 \cdot 3 + \ldots\) can also be expressed as:

\[ = 1 + 2^2 + 3^2 + \ldots \]

### Step 5: Form the Ratio

Now we can form the ratio:

\[ \frac{4(1 + 2^2 + 3^2 + \ldots)}{9(1 + 2^2 + 3^2 + \ldots)} \]

The series cancels out, giving us:

\[ \frac{4}{9} \]

### Step 6: Apply the Power

Now we apply the power of \(\frac{1}{3}\):

\[ \left(\frac{4}{9}\right)^{\frac{1}{3}} = \frac{4^{\frac{1}{3}}}{9^{\frac{1}{3}}} \]

### Step 7: Simplify

This simplifies to:

\[ \frac{2}{3} \]

Thus, the final answer is:

\[ \frac{2}{3} \]

### Final Answer

\(\frac{2}{3}\)