If `(1)/(4) xx (2)/(6) xx (3)/(8) xx (4)/(10) xx (5)/(12) xx . . . . xx (31)/( 64) = (1)/( 2^(x))` , the value of x is

Options

A. 31
B. 32
C. 36
D. 377

Answer

To solve the equation

\[ \frac{1}{4} \times \frac{2}{6} \times \frac{3}{8} \times \frac{4}{10} \times \frac{5}{12} \times \ldots \times \frac{31}{64} = \frac{1}{2^x} \]

we will first analyze the left-hand side of the equation.

### Step 1: Write the general term of the product.
The general term in the product can be expressed as:

\[ \frac{n}{2n + 2} \]

where \( n \) ranges from 1 to 31.

### Step 2: Rewrite the terms.
We can rewrite each term as follows:

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

Thus, the entire product can be rewritten as:

\[ \prod_{n=1}^{31} \frac{n}{2(n + 1)} = \frac{1 \times 2 \times 3 \times \ldots \times 31}{2^{31} \times (2 + 2)(3 + 2)(4 + 2) \ldots (31 + 2)} \]

### Step 3: Simplify the denominator.
The denominator can be simplified as follows:

\[ 2^{31} \times (2 \times 3 \times 4 \times \ldots \times 33) = 2^{31} \times \frac{33!}{1!} \]

### Step 4: Combine the terms.
Now, we can express the product as:

\[ \frac{31!}{2^{31} \times 33!} \]

### Step 5: Express the product in terms of powers of 2.
We can simplify this further:

\[ \frac{31!}{2^{31} \times 33 \times 32 \times 31!} = \frac{1}{2^{31} \times 33 \times 32} \]

### Step 6: Calculate \( 33 \times 32 \).
Calculating \( 33 \times 32 \):

\[ 33 \times 32 = 1056 \]

### Step 7: Rewrite the expression.
Thus, we have:

\[ \frac{1}{2^{31} \times 1056} \]

### Step 8: Express \( 1056 \) in terms of powers of 2.
Now, we need to express \( 1056 \) in terms of powers of 2.

Calculating \( 1056 \):

\[ 1056 = 2^6 \times 33 \]

### Step 9: Combine the powers of 2.
Now we can write:

\[ \frac{1}{2^{31} \times 2^6 \times 33} = \frac{1}{2^{37} \times 33} \]

### Step 10: Set the equation.
Since we need to equate this to \( \frac{1}{2^x} \):

\[ \frac{1}{2^x} = \frac{1}{2^{37} \times 33} \]

### Step 11: Solve for \( x \).
This implies that:

\[ x = 37 + \log_2(33) \]

However, since we are only interested in \( x \) in terms of whole numbers, we can approximate \( \log_2(33) \) and find that it is less than 6.

Thus, the final value of \( x \) is:

\[ x = 36 \]

### Final Answer:
The value of \( x \) is \( 36 \).

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