If `L= lim_(ntooo) (2xx3^(2)xx2^(3)xx3^(4)...xx2^(n-1)xx3^(n))^((1)/((n^(2)+1)))`, then the value of `L^(4)` is _____________. (a) -1/4 (b) 1/2 (c) 1 (d) none of the above

Options

A. `-(1)/(4)`
B. `(1)/(2)`
C. 1
D. 2

Explanation

To solve the limit problem, we start with the expression given:

\[ L = \lim_{n \to \infty} \left( 2 \cdot 3^2 \cdot 2^3 \cdot 3^4 \cdots 2^{n-1} \cdot 3^n \right)^{\frac{1}{n^2 + 1}} \]

### Step 1: Identify the pattern in the product
The product consists of alternating terms of powers of 2 and 3. The odd-indexed terms are powers of 2 and the even-indexed terms are powers of 3.

### Step 2: Separate the powers of 2 and 3
We can rewrite the product as:

\[ L = \lim_{n \to \infty} \left( \left( 2^1 \cdot 2^3 \cdots 2^{n-1} \right) \cdot \left( 3^2 \cdot 3^4 \cdots 3^n \right) \right)^{\frac{1}{n^2 + 1}} \]

### Step 3: Calculate the sum of the powers of 2 and 3
The powers of 2 can be expressed as:

\[ 2^{1 + 3 + 5 + \ldots + (n-1)} \]

The number of terms in the series is \( \frac{n}{2} \) (for even \( n \)). The sum of the first \( k \) odd numbers is \( k^2 \), so:

\[ 1 + 3 + 5 + \ldots + (n-1) = \left( \frac{n}{2} \right)^2 = \frac{n^2}{4} \]

Thus, the contribution from the powers of 2 is:

\[ 2^{\frac{n^2}{4}} \]

For the powers of 3, we have:

\[ 3^{2 + 4 + 6 + \ldots + n} \]

The sum of the first \( k \) even numbers is \( k(k + 1) \), so:

\[ 2 + 4 + 6 + \ldots + n = \frac{n}{2} \cdot \left( \frac{n}{2} + 1 \right) = \frac{n(n + 2)}{4} \]

Thus, the contribution from the powers of 3 is:

\[ 3^{\frac{n(n + 2)}{4}} \]

### Step 4: Combine the results
Now we can combine these results into the limit expression:

\[ L = \lim_{n \to \infty} \left( 2^{\frac{n^2}{4}} \cdot 3^{\frac{n(n + 2)}{4}} \right)^{\frac{1}{n^2 + 1}} \]

### Step 5: Simplify the limit
Taking the limit, we have:

\[ L = \lim_{n \to \infty} \left( 2^{\frac{n^2}{4(n^2 + 1)}} \cdot 3^{\frac{n(n + 2)}{4(n^2 + 1)}} \right) \]

As \( n \to \infty \):

\[ \frac{n^2}{4(n^2 + 1)} \to \frac{1}{4} \]
\[ \frac{n(n + 2)}{4(n^2 + 1)} \to \frac{1}{4} \]

Thus,

\[ L = 2^{\frac{1}{4}} \cdot 3^{\frac{1}{4}} = (2 \cdot 3)^{\frac{1}{4}} = 6^{\frac{1}{4}} \]

### Step 6: Calculate \( L^4 \)
Now, we need to find \( L^4 \):

\[ L^4 = \left( 6^{\frac{1}{4}} \right)^4 = 6 \]

### Final Answer
The value of \( L^4 \) is:

\[ \boxed{6} \]